Research Article Multiresolution Time-Domain Scheme for...

Research ArticleMultiresolution Time-Domain Scheme for Terminal Response ofTwo-Conductor Transmission Lines

Zongliang Tong Lei Sun Ying Li and Jianshu Luo

College of Science National University of Defense Technology Changsha Hunan 410073 China

Correspondence should be addressed to Zongliang Tong tongzlnudteducn

Received 4 January 2016 Revised 20 March 2016 Accepted 18 April 2016

Academic Editor M I Herreros

Copyright copy 2016 Zongliang Tong et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper derives a multiresolution time-domain (MRTD) scheme for the two-conductor lossless transmission line equationsbased on Daubechiesrsquo scaling functions And a method is proposed to generate the scheme at the terminal and near the terminalof the lines The stability and numerical dispersion of this scheme are studied and the proposed scheme shows a better dispersionproperty than the conventional FDTD method Then the MRTD scheme is extended to the two-conductor lossy transmissionline equations The MRTD scheme is implemented with different basis functions for both lossless and lossy transmission linesNumerical results show that theMRTD schemes which use the scaling functions with high vanishingmoment obtainmore accurateresults

1 Introduction

Themultiresolution time-domain (MRTD) scheme proposedby [1] provides an efficient algorithm for electromagnetic fieldcomputation and shows excellent capability to approximateexact solution with low sampling rates However the Battle-Lemarie wavelet function used in [1] is not compact sup-ported which means the iterative equations contain infiniteterms We must cut off the iterative equations in the actualcomputation and this may introduce truncation errors Sodifferent wavelet bases which are compact supported withsome numbers of vanishing moments have been used toimprove this method [2ndash5] This makes a great developmentfor MRTD schemes As a kind of numerical method theMRTD schemes show great advantages in numerical dis-persion properties [6ndash9] meanwhile these schemes needa more rigorous stable condition than the conventionalFDTD method [10] For containing more terms in theiterative equations the terminal conditions or absorbingboundary conditions are more complicated to process inMRTD schemes this disadvantage has limited the applicationof the MRTD scheme To overcome this limitation someworks on the perfect match layer have been made [11ndash13]however other terminal conditions also need to be analyzedspecifically For the transmission lines equations the resistive

terminal conditions could be equivalent as a generalThevenincircuit this paper will solve this kind of terminal condition inthe MRTD scheme

Since the appearance of the telegraph equations studieson transmission lines have had a considerable developmentSeveral equivalent forms of transmission line theory havebeen proposed to describe the influence of the incidentelectromagnetic field to the transmission lines [14ndash16] In [17]the classical theory of the transmission line has been summa-rized and the theory on the high frequency radiation effectsto the transmission lines is introduced In the monograph[18] the multiconductor transmission lines (MTL) theoryhas been comprehensively studied in detail For the two-conductor lossless transmission lines there are several meth-ods which contain the series solution the SPICE solution thetime-domain to frequency-domain (TDFD) transformationmethod and the FDTD method [18] However the MRTDscheme has not been used to calculate the terminal responseof transmission lines In this paper we will derive a MRTDscheme for this problem

In this paper we focus on the calculation of the terminalresponse of two-conductor transmission lines equations byusing MRTD scheme In Section 2 the MRTD scheme isderived based on Daubechiesrsquo scaling functions for thetwo-conductor lossless transmission line equations and for

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8045749 15 pageshttpdxdoiorg10115520168045749

2 Mathematical Problems in Engineering

the resistive terminations the iterative equations for theterminal voltages are derived a method is proposed toupdate the iterative equations which contain some termswhose indices exceed the index range in the MRTD schemeand then the stability and the numerical dispersion arestudied In Section 3 the MRTD scheme is extended tothe two-conductor lossy transmission line In Section 4 thenumerical results are presented on the terminal response ofboth lossless and lossy transmission lines using the MRTDscheme and compared to the FDTDmethod at different spacediscretization numbers and different Courant numbers

2 MRTD Scheme for Two-Conductor LosslessTransmission Lines

21 MRTD Formulation In this section the MRTD schemeis applied to the following scalar transmission lines equationsfor two-conductor lossless lines [18]

120597119881 (119911 119905)

120597119911+ 119897

120597119868 (119911 119905)

120597119905= 0 (1a)

120597119868 (119911 119905)

120597119911+ 119888

120597119881 (119911 119905)

120597119905= 0 (1b)

where 119897 and 119888 are the per-unit-length inductance and capaci-tance respectively

Based on the method outlined in [1] the voltage andcurrent can be expanded as follows

119881 (119911 119905) =

+infin

sum

119896119899=minusinfin

119881119899

119896120601119896(119911) ℎ119899(119905) (2a)

119868 (119911 119905) =

+infin

sum


119868119899+12

119896+12120601119896+12

(119911) ℎ119899+12

(119905) (2b)

where 119881119899

119896and 119868

119899+12

119896+12are the coefficients for the voltages

and currents in terms of scaling functions respectively Theindices 119899 and 119896 are the discrete spatial and temporal indicesrelated to space and time coordinates via 119911 = 119896Δ119911 and119905 = 119899Δ119905 where Δ119911 and Δ119905 represent the spatial and temporaldiscretization intervals in 119911 and 119905 direction The functionℎ119899(119905) is defined as

ℎ119899(119905) = ℎ (

119905

Δ119905minus 119899) (3)

with the rectangular pulse function

ℎ (119905) =

1 for |119905| lt1

21

2for |119905| =

1

2

0 for |119905| gt1

2

(4)

The function 120601119896(119911) is defined as

120601119896(119911) = 120601 (

119911

Δ119911minus 119896) (5)

minus04

minus02

0

02

04

06120601(z)

08

1

12

14

05 1 15 2 25 30z

Figure 1 Daubechiesrsquo scaling function with two vanishingmoments

where 120601(119911) represents Daubechiesrsquo scaling function Figure 1shows Daubechiesrsquo scaling function with two vanishingmoments

For deriving theMRTD scheme for (1a) and (1b) we needthe following integrals

int

+infin

minusinfin

ℎ119899(119905) ℎ1198991015840 (119905) 119889119905 = 120575

1198991198991015840Δ119905

int

+infin

minusinfin

ℎ119899(119905)

120597ℎ1198991015840+12

(119905)

120597119905119889119905 = 120575

1198991198991015840 minus 1205751198991198991015840+1

(6)

where 1205751198991198991015840 represents the Kronecker symbol Consider

int

+infin

minusinfin

120601119896(119911) 1206011198961015840 (119911) 119889119911 = 120575

1198961198961015840Δ119911 (7)

int

+infin

minusinfin

120601119899(119911)

1205971206011198991015840+12

(119911)

120597119911119889119911 =

119871119878minus1

sum

119894=minus119871119878

119886 (119894) 120575119899+1198941198991015840 (8)

where 119871119878denotes the effective support size of the basis func-

tions The coefficients 119886(119894) are called connection coefficientsand can be calculated by (9) Taking Daubechiesrsquo scalingfunctions as the basis functions Table 1 shows 119886(119894) for 0 le

119894 le 119871119878minus 1 which are zeros for 119894 gt 119871

119878minus 1 and for 119894 lt 0 it can

be obtained by the symmetry relation 119886(minus1 minus 119894) = minus119886(119894)

119886 (119894) =1

120587int

infin

0

10038161003816100381610038161003816 (120582)

10038161003816100381610038161003816

2

120582 sin 120582 (119894 +1

2) 119889120582 (9)

where (120582) represents the Fourier transform of 120601(119911)Daubechiesrsquo scaling functions satisfy the shifted interpo-

lation property [19]

120601 (119894 + 1198721) = 1205751198940

(10)

for 119894 integer where1198721= int+infin

minusinfin119911120601(119911)119889119911 is the first moment of

the scaling functions and the values of1198721are listed in Table 1

Following the theory in [3] and making use of (10) (5) ismodified to

120601119896(119911) = 120601 (

119911

Δ119911minus 119896 + 119872

1) (11)

Mathematical Problems in Engineering 3

Table 1 Connection coefficients 119886(119894) and the first-order moments1198721of Daubechiesrsquo scaling functions

119894 1198632

1198633

1198634

0 12291666667 12918129281 131103407731 minus00937500000 minus01371343465 minus015601001102 00104166667 00287617728 004199574603 minus00034701413 minus000865432364 00000080265 000083086955 000001089996 000000000411198721

06339743121 08174005815 10053923835119902max 07500 06844 06585

In spite of the support of the scaling functions [20] single-point sampling of the total voltages and currents can betaken at integer points with negligible error Taking voltageat spatial point 119896Δ119911 and at time 119899Δ119905 we obtain

119881 (119896Δ119911 119899Δ119905)

= ∬

+infin

minusinfin

119881 (119911 119905) 120575 (119911

Δ119911minus 119896) 120575 (

119905

Δ119905minus 119899) 119889119911 119889119905

= 119881119899

119896

(12)

where 120575 is the Dirac delta function Equation (12) means thevoltage value at each integer point is equal to the coefficientThe current values have the same character at each halfinteger point Therefore we will use 119881

119899

119896and 119868119899+12

119896+12directly to

represent the voltage at the point (119896Δ119911 119899Δ119905) and the currentat the point ((119896 + 12)Δ119911 (119899 + 12)Δ119905) in this paper

The modified 120601119896(119911) in (11) also satisfy integrals (7) and

(8) Applying the Galerkin technique to (1a) and (1b) we canobtain the following iterative equations for the voltages andcurrents

119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (13a)

119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (13b)

22 Terminal Iterative Equations for Resistive Load in MRTDScheme We will consider the terminal conditions for thetwo-conductor lossless transmission lines equations in thissection Equations (1a) and (1b) are homogeneous linearequations we need to add the terminal conditions to obtainthe unique solution

Considering the two-conductor lines shown in Figure 2we assume the length of the total line is 119871 and the resistiveloads are 119877

119878and 119877

119871 The line is divided uniformly into

NDZ segments with the space interval Δ119911 and the totalsolution time is divided into NDT steps with the uniformtime interval Δ119905 Similar to the conventional FDTD wewill calculate the interlace voltages 119881

119899

0 119881119899

1 119881

119899

NDZ andcurrents 119868

119899+12

12 119868119899+12

32 119868

119899+12

NDZminus12 in both space domain

+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

z

VS(t)

RS

Figure 2 A two-conductor line in time-domain

and time-domain as shown in (13a) and (13b) for 119899 =

1 2 NDTFor the resistive terminations we note the voltage at the

source (119911 = 0) as 119881119878(119905) and the current at the source as 119868

119878(119905)

the external voltage at the load (119911 = 119871) as 119881119871(119905) and the

current at the load as 119868119871(119905) The discrete voltages and currents

at the source are denoted as 119881119899119878

equiv 119881119878(119899Δ119905) and 119868

119899

119878equiv 119868119878(119899Δ119905)

and the discrete voltages and currents at the load are denotedas 119881119899

119871equiv 119881119871(119899Δ119905) and 119868

119899

119871equiv 119868119871(119899Δ119905) then the terminal

characterizations could be written in terms of a generalizedThevenin equivalent as

119881119899

0= 119881119899

119878minus 119877119878119868119899

119878(14a)

119881119899

NDZ = 119881119899

119871+ 119877119871119868119899

119871 (14b)

Equations (14a) and (14b) denote the discretization terminalconditions for the case of resistive terminations so we needto introduce these conditions to the iterative equations (13a)and (13b) to obtain the numerical solution

Notice that in the iterative equations (13a) and (13b) notonly the iterative equations of the terminal voltages119881119899+1

0and

119881119899+1

NDZ should be derived and the iterative equations of voltagesand currents ldquonearrdquo the terminals also need to be updatedThe voltages and currents ldquonearrdquo the terminals we mean arethe voltages 119881

119899+1

119894and 119881

119899

NDZminus119894 for 119894 = 1 2 119871119878minus 1and

the currents 119868119899+12

119894+12and 119868

119899+12

NDZminus119894+12 for 119894 = 0 1 119871119878minus 2

All of these voltages and currents contain some terms thatexceed the index range in iterative equations (13a) and (13b)Figure 3 shows the discretization of the terminal voltages andthe voltages and currents near the terminal

We will derive the MRTD scheme at the terminal firstlyFor updating the iterative equations for the terminal voltageswe need to decompose iterative equations ((13a) (13b)) Sincethe coefficients 119886(119894) satisfy the following relation [4]

119871119878minus1

sum

119894=0

(2119894 + 1) 119886 (119894) = 1 (15)

substituting (15) into (13a) we can obtain119871119878minus1

sum

119894=0

119886 (119894) (2119894 + 1)119881119899+1

119896=

119871119878minus1

sum

119894=0

119886 (119894) (2119894 + 1)119881119899

119896

minus

119871119878minus1

sum

119894=0

Δ119905

(2119894 + 1) Δ119911119888minus1

[119886 (119894) (2119894 + 1)

sdot (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12)]

(16)


Iminus12 V0 I12 V1

minusΔz

2

Δz

2

+

minus

+

minus

z = 0 Δz

VNDZminus1 INDZminus12 VNDZ INDZ+12

IL

(NDZ minus 1)Δz NDZΔz z

(NDZ minus12)Δz (NDZ +

12)Δz

IS

Figure 3 Discretizing the terminal voltages and currents

Considering the corresponding terms with 119894 we candecompose (13a) as [21]

119886 (119894) (2119894 + 1)119881119899+1

119896

= 119886 (119894) (2119894 + 1) 119881119899

119896

minus 119886 (119894) (2119894 + 1)Δ119905

(2119894 + 1) Δ119911119888minus1

(119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12)

(17)

for 119894 = 0 1 119871119878

minus 1 Equation (13b) could make theanalogous decompositionWe could view theMRTD schemefor two-conductor transmission lines as the weighted meanof the conventional FDTDmethod with spatial discretizationstep (2119894 + 1)Δ119911 for 119894 = 0 1 119871

119878minus 1 and the weighting

coefficient for each term is (2119894+1)119886(119894) Besides for theMRTDscheme whose coefficients 119886(119894) satisfy relationship (15) theanalogous decomposition could be made This relationshipbetween the MRTD scheme and the conventional FDTDmethod is useful for us to update the iterative equations

Taking 119881119899+1

0as an example to derive the iterative equa-

tions at the terminal

119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119894+12minus 119868119899+12

minus119894minus12) (18)

Following steps of (16) and (17) we can decompose (18)as

119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

12minus 119868119899+12

minus12) (19a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

32minus 119868119899+12

minus32) (19b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

minus119871119878+12

)

(19c)

Here we could view each equation in (19a) (19b) and(19c) as a central difference scheme (19a) is the centraldifference scheme related to points 119911 = minusΔ1199112 and 119911 =

Δ1199112 (19b) is the central difference scheme related to points119911 = minus3Δ1199112 and 119911 = 3Δ1199112 and so on but the terms119868119899+12

minus12 119868119899+12

minus32 119868

119899+12

minus119871119878+12

whose subscripts exceed the indexrange make (19a) (19b) and (19c) out of work So we needto make some update for the iterative equations Using theforward difference scheme to replace the central differencescheme we change the difference points in (19a) to become119911 = 0 and 119911 = Δ1199112 and change the difference points in(19b) to become 119911 = 0 and 119911 = 3Δ1199112 the others followthe same step Keeping the weighting coefficient unchangedin each equation we can obtain

119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ1199112119888minus1

(119868119899+12

12minus 119868119899+12

119878) (20a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

(3Δ119911) 2119888minus1

(119868119899+12

32minus 119868119899+12

119878) (20b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ1199112

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

119878)

(20c)

where the terminal current 119868119899+12119878

= (119868119899

119878+ 119868119899+1

119878)2 and 119868

119899

119878can

be derived from (14a)

119868119899

119878=

(119881119899

119878minus 119881119899

0)

119877119878

(21)

Summing up all the equations in (20a) (20b) and (20c)

119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

119894+12minus 119868119899+12

119878) (22)


Substituting (21) into (22) we can obtain the iterativeequation at the source

119881119899+1

0= (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(Δ119911

Δ119905119888119877119878minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0minus 2119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12

+

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(23)

With the same steps we can obtain the iterative equationat the load

119881119899+1

NDZ = (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(Δ119911

Δ119905119888119877119871minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 2119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12 +

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(24)

After deriving the iterative equations at the terminal wewill put forward a truncation method to update iterativeequations which contain some terms whose indices exceedthe index range in the MRTD scheme

Taking 119881119899+1

119896as an example for 119896 = 1 2 119871

119878minus 1

decomposing (13a)

119886 (0) 119881119899+1

119896= 119886 (0) 119881

119899

119896minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

119896+12minus 119868119899+12

119896minus12) (25a)

3119886 (1) 119881119899+1

119896= 3119886 (1) 119881

119899

119896minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

119896+32minus 119868119899+12

119896minus32) (25b)

(2119896 minus 1) 119886 (119896 minus 1)119881119899+1

119896minus1= (2119896 minus 1) 119886 (119896 minus 1)119881

119899

119896minus1

minus (2119896 minus 1) 119886 (119896 minus 1)Δ119905

(2119896 minus 1) Δ119911119888minus1

(119868119899+12

2119896minus12minus 119868119899+12

12)

(25c)

(2119896 + 1) 119886 (119896) 119881119899+1

119896= (2119896 + 1) 119886 (119896) 119881

119899

119896minus (2119896 + 1) 119886 (119896)

sdotΔ119905

(2119896 + 1) Δ119911119888minus1

(119868119899+12

2119896+12minus 119868119899+12

minus12)

(25d)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

119871119878minus1

= (2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899

119871119878minus1

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119896+119871119878minus32

minus 119868119899+12

119896minus119871119878+12

)

(25e)

Noticing the first 119896 terms in (25a) (25b) (25c) (25d)and (25e) there is no term exceeding the index range in

each equation Meanwhile the equations which contain theexceeding indices terms are all appearing in the rest of 119871

119878minus 119896

terms As mentioned before we can view the MRTD schemeas the weightedmean of the conventional FDTDmethod but(25a) (25b) (25c) (25d) and (25e) show that the last 119871

119878minus 119896

equations are unavailable for forming the iterative equationsinMRTD scheme To solve this problem wemake truncationhere We update the iterative equation of 119881119899+1

119896by using the

summation of the first 119896 terms in (25a) (25b) (25c) (25d)and (25e) that means we use the weighted mean of the first 119896to approximate the summation of all the 119871

119878terms

Summing up the first 119896 terms in (25a) (25b) (25c) (25d)and (25e) we can obtain the modified iterative equations

119881119899+1

119896= 119881119899

119896minus (

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(26)

for 119896 = 1 2 119871119878minus 1

Using the same method we can obtain the modifiediterative equations near the load

119881119899+1

119896= 119881119899

119896minus (

NDZminus119896minus1sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(27)

for 119896 = NDZ minus 119871119878+ 1NDZ minus 119871

119878+ 2 NDZ minus 1

The voltages at the interior points are determined from(13a)

119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (28)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878

For the iterative equation of the current there is a littledifference from the voltagersquos As shown in Figure 3 theinterlace currents appear at the half integer points whichmeans all the currents are located at the interior points Sowe only need to modify the currents near the terminalsFollowing the same steps of the derivation of voltages iterativeequations near the terminal we could obtain the currentiterative equations near the terminals

For the current iterative equations near the source

119868119899+12

119896+12= 119868119899minus12

119896+12minus (

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(29)

for 119896 = 0 1 119871119878minus 2


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

LL

z

VS(t)

RS

Figure 4 A two-conductor line with an inductive resistance

For the current iterative equations near the load

119868119899+12

119896+12= 119868119899minus12

119896+12minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(30)


119878+ 2 NDZ minus 1

The currents at the interior points are determined from(13b)

119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (31)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878

23 Terminal Iterative Equations for Inductive Resistance inMRTD Scheme Since we have discussed the resistive loadin Section 22 we will consider a more complicated terminalload which consisted of a resistance and inductance shownin Figure 4 The resistance and the inductance at the load arenoted as 119877

119871and 119871

119871 respectively

Keeping the source as a resistive terminal and changingthe load including inductance we note the voltage at thesource (119911 = 0) as 119881

119878(119905) and the current at the source as 119868

119878(119905)


current at the load as 119868119871(119905) The terminal conditions could be

written as follows

1198810(119905) = 119881

119878(119905) minus 119877

119878119868119878(119905) (32a)

119881NDZ (119905) = 119881119871(119905) + 119877

119871119868119871(119905) + 119871

119871

119889119868119871(119905)

119889119905 (32b)

Expanding 1198810(119905) 1198680(119905) 119881119871(119905) and 119868

119871(119905) as (2a) and (2b)

and sampling them at the time discreting point (119899 + 12)Δ119905the terminal conditions become

119881119899+1

0+ 119881119899

0

2=

119881119899+1

119878+ 119881119899

119878

2minus 119877119878119868119899+12

119878

(33a)

119881119899+1

NDZ + 119881119899

NDZ2

=119881119899+1

119871+ 119881119899

119871

2+ 119877119871119868119899+12

119871

+119871119871

2Δ119905(119868119899+32

119871minus 119868119899minus12

119871)

(33b)

It could be seen from Section 22 that the change ofthe terminal condition only affects the iterative equationsat the terminals so we just need to consider two iterativeequations Since we keep the source as a resistive terminationthe iterative equation at the source should be the same as (23)Actually if we substitute (33a) into (22) we could obtain (23)So the only iterative equation we should derive is located atthe load If we set 119871

119871= 0 in (33b) the terminal condition at

the load will have the analogous form with the source whichmeans the load with a resistance and inductance degeneratesto be a resistance For those 119871

119871= 0 transforming (33b) as

119868119899+32

119871= 119868119899minus12

119871+

Δ119905

119871119871

((119881119899+1

NDZ + 119881119899

NDZ) minus (119881119899+1

119871+ 119881119899

119871)

minus 2119877119871119868119899+12

119871)

(34)

Following the steps we get the iterative equation of 119881119899+10

in Section 22 we can obtain the iterative equation at the load

119881119899+1

NDZ = 119881119899

NDZ

minusΔ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(35)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ) minus (119881119899

119871+ 119881119899minus1

119871)

minus 2119877119871119868119899minus12

119871)

(36)

for 119899 = 2 3 NDT

24 Stability Analysis For the purpose of stability analysis(13a) and (13b) can be rewritten as

119881119899+1

119896minus 119881119899

119896

Δ119905= minus

1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (37a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= minus

1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (37b)

Following the procedures in [22] the finite-differenceapproximations of the timederivations on the left side of (37a)and (37b) can be written as an eigenvalue problem

119881119899+1

119896minus 119881119899

119896

Δ119905= 120582119881119899+12

119896

(38a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12

(38b)

In order to avoid instability during normal time steppingthe imaginary part of 120582 must satisfy

|Im (120582)| le2

Δ119905 (39)


2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40

Samples per wavelength

Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)

Figure 5 Phase error (in degrees) for different MRTD schemes The Courant numbers are 119902 = 05 for (a) and 119902 = 01 for (b)

As we consider the lossless two-conductor transmissionlines in (1a) and (1b) the transient values of voltages andcurrents distributed in space can be Fourier transformedwithrespect to 119896-coordinates to provide a spectrum of sinusoidalmodes Assuming an eigenmode of the spectral domain with119896119911 the voltages and currents can be written as

119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)

where 119881119870represents the voltage at point 119870Δ119911 119868

119870+12repre-

sents the current at point (119870+12)Δ119911 and 1198811199110

and 1198681199110

are theamplitudes of the voltages and currents

Substituting (40a) and (40b) into (38a) and (38b) and(37a) and (37b) we obtain

1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)

In (41) 120582 is pure imaginary and

|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)

where V = 1radic119897119888 is the velocity of the wave along with thelines

Numerical stability is maintained for every spatial modeonly when the range of eigenvalues given by (42) is containedentirely within the stable range of time-difference eigenvaluesgiven by (39) Since both ranges are symmetrical around zeroit is adequate to set the upper bound of (42) to be smaller orequal to (39) we can obtain the stability condition

VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)

Noting the Courant number

119902 =VΔ119905

Δ119911(44)

the maximum values of 119902 required by a stable algorithmwhich are listed in Table 1 as 119902max can be calculated from theconnection coefficients

25 Dispersion Analysis To calculate the numerical disper-sion substituting a time harmonic trial solution into (37a)and (37b) we can obtain

1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)

where120596 is thewave angular frequency and 119896119911is the numerical

wave numberUsing the number of cells per wavelength 119899

119897= 120582REALΔ119911

and the wave number 119896119911

= (2120587)120582NUM we obtain thedispersion relationship

1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)

where 119906 = 120582REAL120582NUM is the ratio between the theoreticaland numerical wavelength

The value of 119906 could be computed by Newton iterativemethod and using the formula |(119906 minus 1) times 360| calculates thephase error in degrees Figure 5 shows the phase errors ofdifferent MRTD schemes versus the samples per wavelengthCompared with the conventional FDTD method at thesame sampling numbers the MRTD schemes show betternumerical dispersion properties than FDTD method whichmeans the discretization error of the MRTD schemes issmaller than the FDTD method


3 MRTD Scheme for Two-Conductor LossyTransmission Lines

In this section we will extend the MRTD scheme to the two-conductor lossy transmission lines

31 MRTD Formulation For the lossy case the two-conductor transmission line equations become

120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)

where 119903 119897 119892 and 119888 are the per-unit-length resistance induc-tance conductance and capacitance respectively

Compared to the lossless transmission lines the lossycase must consider the resistance losses along the lines andthe losses in the medium However the steps to obtain theMRTD scheme are the same Similar to the lossless casewe firstly extend the voltage and current with Daubechiesrsquoscaling functions and the rectangular function and use 119881

119899

119896

and 119868119899+12

119896+12representing the voltage at the point (119896Δ119911 119899Δ119905) and

the current at the point ((119896+12)Δ119911 (119899+12)Δ119905) respectivelyand Δ119911 and Δ119905 represent the spatial and temporal discretiza-tion intervals By applying the Galerkin technique we canobtain the iterative equations for the lossy transmission linesequations

119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)

where 119886(119894) is the connection coefficient and 1198751 1198752 1198761 and

1198762are constants

1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)

The differences between (48a) and (48b) and (13a) and(13b) are the coefficients of terms in the iterative equationsthat are caused by the unit-per-length resistance 119903 andinductance 119892 If we set 119903 = 0 and 119892 = 0 the coefficients 119875

1

1198752and 119876

1 1198762are all equal to 1 and the iterative equations

(48a) and (48b) will degenerate to iterative equations (13a)and (13b)

For the lossy case we should also modify the iterativeequations at the terminal and near the terminal of the linesConsidering the two-conductor lines shown in Figure 2 weassume the length of the total line is 119871 and the resistiveloads are 119877

119878and 119877

119871 The line is divided uniformly into NDZ

segments with the space interval Δ119911 and the total solutiontime is divided intoNDT steps with the uniform time intervalΔ119905 Following the same steps in Section 22 we could obtainthe modified iterative equations

For the voltage iterative equation at the source

119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)

For the voltage iterative equation at the load

119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)

For the voltage iterative equations near the source

119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1

For the voltage iterative equations near the load

119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878

Besides for the two-conductor line shown in Figure 4 wecan obtain the iterative equation at the load

119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT

32 Stability Analysis To study the stability of the MRTDscheme for lossy case we need to make some changesRewrite (48a) and (48b) as follows

119875minus1

2119881119899+1

119896minus 119875minus1

21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)

The finite-difference approximations of the time deriva-tions on the left side of (61a) and (61b) are different from theleft side of (37a) and (37b) but we can also write them as aneigenvalue problem

119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)

In order to avoid instability during normal time steppingthe imaginary part of 120582 also must satisfy

|Im (120582)| le2

Δ119905 (63)

Then following the steps in Section 24 using the Fouriertransform expand the transient values of voltages and cur-rents distributed in space We can also obtain the samestability condition as (43)

VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)

However the dispersion analysis for the iterative equa-tions of the lossy transmission lines is quite different fromthe lossless case Since 119875

minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that

is the coefficient of 119881119899+1119896

is not equal to the coefficient of 119881119899119896

and the coefficient of 119868119899+12119896+12

is not equal to the coefficient of119868119899minus12

119896+12on the left side of (61a) and (61b) if we substitute a time

harmonic into the iterative equations into (61a) and (61b)there are some other terms that could affect the ratio betweenthe theoretical and numerical wavelength except samples perwavelength so we could not obtain a brief dispersion relationas (45) However the numerical results in Section 42 showthat the MRTD method could obtain a more accurate resultthan the FDTD method at the same space interval and timeinterval


t

A

120591f120591r

VS(t)

Figure 6 Trapezoidal pulse with the rise time 120591119903and fall time 120591

119891

4 Numerical Result

41 Lossless Transmission Lines In this section the two-conductor lossless transmission lines as shown in Figure 2 areconsidered to calculate the terminal voltages The length ofthe lines is 119871 = 400m and the per-unit-length capacitanceand inductance are 119888 = 100 pFm and 119897 = 025 120583Hmrespectively This corresponds to RG58U coaxial cable [18]The characteristic impedance of the line is119885

119862= radic119897119888 = 50Ω

and the velocity of propagation is V = 1radic119897119888 = 200m120583sTheterminal load is 100Ω and the source resistance is 50Ω

We will use a trapezoidal pulse as shown in Figure 6 asthe source voltage whose initial value is 0V and amplitudeis 119860 = 30V total time is 10 120583s rise time is 120591

119903= 1 120583s and

fall time is 120591119891

= 1 120583s respectively The bandwidth of thepulse is approximate to BW = 1120591

119903 so the segment length

Δ119911 should be electrically short at this frequency requiringΔ119911 le (110)(V119891max) (119891max = 1120591

119903) [18] As we take the

rise time of the trapezoidal pulse to be 1 120583s the approximatebandwidth of the pulse is 1MHz and the maximum sectionlength of Δ119911 should be less than 20m noting Δ119911max = 20Since we divide the line into NDZ segments uniformly NDZshould be greater than 119871Δ119911max = 20 The time step will becalculated by (44) with different Courant number 119902

Under the conditions of the space discretization numberNDZ = 20 and the Courant number 119902 = 05 we calculatethe terminal voltages using the MRTD schemes with 119863

2

1198633 and 119863

4waveletsrsquo scaling functions We also calculate

the terminal voltages by the FDTD method under the sameconditions Figure 7 shows the numerical results by differentmethods and 119863

119894-MRTD represents the MRTD scheme using

Daubechiesrsquo scaling functions with 119894 vanishing moment asbasis functions where 119894 = 2 3 4

For the time-dependent discrete terminal voltages therelative error is defined as follows [23]

120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)

Here 119909(119894) represents the numerical results of FDTD andMRTD at each time discretization point and (119894) is thenumerical result of series solution at each time discretizationpoint which can be regarded as the exact solution [18] Taking(119894) as the exact result we calculate the relative errors ofFDTD method and MRTD scheme

minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)

Figure 7 Terminal voltage calculated by different numerical meth-ods The space discretization number and the Courant number areNDZ = 20 and 119902 = 05

Table 2 Relative errors and runtime of different methods (NDZ =20 119902 = 05)

Analysis scheme Relative error Runtime (s)FDTD 108 000111198632-MRTD 193 00050

1198633-MRTD 066 00054

1198634-MRTD 062 00058

Table 2 shows the relative errors and runtime for differentschemes Since the iterative equations in MRTD schemescontain more terms than the conventional FDTD methodthe MRTD schemes expend more runtime When we use1198632-MRTD the numerical result shows a larger relative error

The reason is that the vanishing moment of the 1198632waveletrsquos

scaling function is not high enough For the waveletsrsquo scalingfunctions whose vanishing moment is high enough like 119863

3

wavelet and 1198634wavelet the numerical results show smaller

relative errorsFigure 8 shows the relative errors versus the space dis-

cretization numbersThe space interval will decrease with theincrease of the space discretization number We can see fromthe figures that the relative errors for119863

2-MRTD increasewith

the increase of NDZ And for1198633-MRTD and119863

4-MRTD the

relative errors decrease with the increase of NDZ and showa little smaller relative error than the conventional FDTD Itmeanswe can get amore accurate result at same space intervaland time interval by 119863

3-MRTD and 119863

4-MRTD

Figure 9 describes the relative errors versus the Courantnumbers The time interval will decrease with the decreaseof the Courant number The results show that with thedecrease of the Courant numbers the relative errors arealmost unchanged for the conventional FDTDmethod whilethe relative errors are quite different with different Courant


20 30 40 50 60 70 80 90 1000

05

1

15

2

25

Space discretization number

Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)

Figure 8 Relative error for different discretization numbers The Courant numbers are 119902 = 05 for (a) and 119902 = 01 for (b)

02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)

Figure 9 Relative errors for different Courant numbers The space discretization numbers are NDZ = 20 for (a) and NDZ = 50 for (b)

numbers for all the three MRTD schemes That means thechoice of the Courant number significantly affects the relativeerrors of theMRTD schemes andwe could choose the properCourant number to optimize the MRTD schemes

To validate the stability of the MRTD schemes weincrease the initial value of the pulse to 20V and keepother parameters unchanged Figure 10 shows the relativeerrors versus the pace discretization numbers It can be seenfrom the figures that the 119863

2-MRTD scheme shows a larger

relative error and the relative errors of 1198633-MRTD and 119863

4-

MRTD schemes are smaller than the conventional FDTDmethod

The numerical results for the lossless transmission linesalso show that the 119863

2-MRTD does not perform better than

the FDTD method meanwhile 1198633-MRTD and 119863

4-MRTD

schemes show better quality in accuracy and stability Thereason for this phenomenon is that the scaling function of the1198632wavelet does not have enough high vanishing moment

When we use Daubechiesrsquo scaling functions to expand thevoltages and currents in the two-conductor transmissionlines equations the vanishing moment decides the accuracyof the approximation The scaling functions with high van-ishing moments could approximate voltages and currentsmore accurately however the scaling functions with low


14

16

18

2

22

24

26

28

3

32

34

20 30 40 50 60 70 80 90 100Space discretization number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)Figure 10 Relative error for different space discretization numbers for the initial value of the trapezoidal pulse to be 20V The Courantnumbers are 119902 = 05 for (a) and 119902 = 01 for (b)

+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12

(b)Figure 11 A lossy printed circuit board (a) line dimensions and terminations and (b) cross-sectional dimensions

vanishing moments like 1198632waveletrsquos may introduce a larger

error So we can see fromFigure 5 that1198632-MRTDhas a better

numerical dispersion property than the FDTD method butit gets larger relative errors in numerical computation Forthe scaling functions which have high enough vanishingmoments like 119863

3wavelet and 119863

4wavelet the numerical

results show smaller relative errors and are in agreementwith the dispersion analysis However a high vanishingmoment for scaling function may increase the computationcomplexity in MRTD schemes So the vanishing momentsof the waveletrsquos scaling functions have a great effect on theaccuracy of the MRTD scheme it is necessary to choose ascaling function with proper vanishingmoment when we usethe MRTD scheme for the numerical computation

42 Lossy Transmission Lines In this section we will con-sider the lossy two-conductor transmission lines shown inFigure 11 Two conductors of rectangular cross section ofwidth 119908 = 20 120583m and thickness 119905 = 10 120583m are separatedby 119904 = 20 120583m and placed on one side of a silicon substrate(120576119903

= 12) of thickness ℎ = 100 120583m the total line length is119871 = 20 cm The near end is a source with a 119877

119878=

50Ω resistance and the far end is a load with a 119877119871

= 50Ω

resistance and 119871119871

= 05 120583H inductance in series The per-unit-length inductance and capacitance were computed as 119897 =0805969 120583Hmand 119888 = 882488 pFmThis gives a velocity of

t05nsO

1V

VS(t)

Figure 12 Representation of the source voltage waveform

V = 118573 times 108ms and a one-way time delay of

119879119863

= 168672 ns which gives an effective dielectric constantof (1205761015840119903

= 64) and characteristic impedance of 119885119862

=

95566Ω The per-unit-length dc resistance is computed as119903 = 1(120590119908119905) = 86207Ωm [18] Dielectric loss is notincluded in these calculations which means the per-unit-length conductance is 119892 = 0

The source is a ramp function as shown in Figure 12 theinitial value of the ramp function is 0V and the amplitude is119860 = 1V with a rise time of 120591

119903= 05 ns The total computing

time is 20 ns The bandwidth of the source is approximate toBW = 1120591

119903= 2GHz The space discretization step for the

MRTD was chosen to be 12058210 so the space discretization


0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(b)Figure 13 Terminal voltages for lossy transmission lines (a) for the near end and (b) for the far end The space discretization number isNDZ = 40 and the Courant number is 119902 = 05

Table 3 Relative errors and runtime of different methods 1198810

represents the near end voltage and119881119871represents the far end voltage

(NDZ = 40 119902 = 05)

Analysisscheme

Relative errorfor 1198810


Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275

number NDZ should be greater than 34 The time step willalso be calculated by (44) with different Courant numbers

We calculate the near end voltage and the far end voltageof the lossy PCB with space discretization number NDZ =40 and the Courant number 119902 = 05 Figure 13 shows thecomputing results Since 119863

2-MRTD may introduce a larger

error in the computation as shown in Section 41 we use 1198633-

MRTD and 1198634-MRTD to compute the terminal voltages

Here we choose the time-domain to frequency-domaintransformation method (TDFD) which is a straightforwardadaptation of a common analysis technique for lumpedlinear circuits and systems [18] to validate the computingresults of MRTD schemes and FDTD method Table 3 showsthe relative errors and the runtime of the MRTD schemesand FDTDmethod It can be seen thatMRTD schemes spendmuch time to obtain more accurate results

Figure 14 describes the relative errors versus the spacediscretization numbers For both119863

3-MRTD and119863

4-MRTD

the MRTD schemes show a little smaller relative error thanthe conventional FDTD That means the MRTD schemescould obtain a more accurate solution under the same timeinterval and space interval This is because 119863

3-MRTD and

1198634-MRTD have better dispersion property than the FDTD

method and the scaling functions of 1198633wavelet and 119863

4

wavelet have high enough vanishing momentsFigure 15 shows the relative errors versus the Courant

numbers With the decrease of the Courant number therelative errors of the FDTD method increase because thetime-domain is oversampled for the FDTD method How-ever the MRTD schemes perform decreasing relative errorsThatmeans theMRTD schemes could obtain amore accurateresult with a smaller time interval And even when the FDTDmethod is oversampled in the time-domain the MRTDschemes perform well

5 Conclusion

In this paper we derived the MRTD scheme for the two-conductor transmission lines and studied the stability and thenumerical dispersion of this scheme By viewing the MRTDschemes as the weighted mean of the conventional FDTDmethod at different space interval we derived the iterativeequations for the terminal voltages when the terminals arepure resistive and a method is proposed to update theiterative equations which contain some terms whose indicesexceed the index range in theMRTD scheme Using the samemethod we derived the iterative equation for the inductiveload Then we extended the MRTD scheme to the lossytransmission lines Using different waveletsrsquo scaling functionsas basis functions the MRTD schemes are implemented forboth lossless case and lossy case and the numerical results arecompared to the conventional FDTDmethodThe numericalresults show the MRTD schemes need more runtime toobtain more accurate results And the vanishing momentof the waveletrsquos scaling functions will significantly affect thequality of the MRTD scheme using a scaling function with aproper vanishingmoment as basis function inMRTD schemecould obtain a more accurate result


005

01

015

02

025

03

40 50 60 70 80 90 100Space discretization number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)

Figure 14 Relative error for the terminal voltages with different space discretization numbers (a) for the near end and (b) for the far endThe Courant number is 119902 = 05

005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)Figure 15 Relative errors for the terminal voltages with different Courant numbers (a) for the near end and (b) for the far end The spacediscretization number is NDZ = 40

Competing Interests

The authors declare that they have no competing inter-ests

Acknowledgments

The authors would like to thank Yinkun Wang for his com-ments on the paperThis work was supported by the NationalNatural Science Foundation of China (no 11271370)

References

[1] M Krumpholz and L P B Katehi ldquoMRTD new time-domainschemes based on multiresolution analysisrdquo IEEE TransactionsonMicrowaveTheory and Techniques vol 44 no 4 pp 555ndash5711996

[2] M Fujii and W J R Hoefer ldquoA three-dimensional Harr-wavelet-based multi-resolution analysis similar to the 3-DFDTD method-derivation and applicationrdquo The IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2463ndash2475 1998


[3] Y W Cheong Y M Lee K H Ra J G Kang and C C ShinldquoWavelet-Galerkin scheme of time-dependent inhomogeneouselectromagnetic problemsrdquo IEEEMicrowave andWireless Com-ponents Letters vol 9 no 8 pp 297ndash299 1999

[4] T Dogaru and L Carin ldquoMultiresolution time-domain usingCDF biorthogonal waveletsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 49 no 5 pp 902ndash912 2001

[5] X CWei E P Li andCH Liang ldquoA newMRTD scheme basedon Coifman scaling functions for the solution of scatteringproblemsrdquo IEEE Microwave and Wireless Components Lettersvol 12 no 10 pp 392ndash394 2002

[6] E M Tentzeris R L Robertson J F Harvey and L P BKatehi ldquoStability and dispersion analysis of battle-lemarie-based MRTD schemesrdquo The IEEE Transactions on MicrowaveTheory and Techniques vol 47 no 7 pp 1004ndash1013 1999

[7] S Grivet-Talocia ldquoOn the accuracy of haar-based multireso-lution time-domain schemesrdquo IEEE Microwave and WirelessComponents Letters vol 10 no 10 pp 397ndash399 2000

[8] M Fujii andW J RHoefer ldquoDispersion of time domainwaveletgalerkin method based on Daubechiesrsquo compactly supportedscaling functionswith three and four vanishingmomentsrdquo IEEEMicrowave and Wireless Components Letters vol 10 no 4 pp125ndash127 2000

[9] A Alighanbari and C D Sarris ldquoDispersion properties andapplications of the Coifman scaling function based S-MRTDrdquoIEEE Transactions on Antennas and Propagation vol 54 no 8pp 2316ndash2325 2006

[10] K L Shlager and J B Schneider ldquoComparison of the dis-persion properties of several low-dispersion finite-differencetime-domain algorithmsrdquo IEEE Transactions on Antennas andPropagation vol 51 no 3 pp 642ndash653 2003

[11] E M Tentzeris R L Robertson J F Harvey and L P B KatehildquoPML absorbing boundary conditions for the characterizationof open microwave circuit components using multiresolu-tion time-domain techniques (MRTD)rdquo IEEE Transactions onAntennas and Propagation vol 47 no 11 pp 1709ndash1715 1999

[12] Q S Cao Y C Chen and R Mittra ldquoMultiple image technique(MIT) and anisotropic perfectly matched layer (APML) inimplementation of MRTD scheme for boundary truncations ofmicrowave structuresrdquo IEEE Transactions onMicrowaveTheoryand Techniques vol 50 no 6 pp 1578ndash1589 2002

[13] YW Liu YW Chen B Chen and X Xu ldquoA cylindrical MRTDalgorithm with PML and quasi-PMLrdquo IEEE Transactions onMicrowave Theory and Techniques vol 61 no 3 pp 1006ndash10172013

[14] C D Taylor R S Satterwhite and C W Harrison ldquoThe re-sponse of a terminated two-wire transmission line excited bya nonuniform electromagnetic fieldrdquo IEEE Transactions onAntennas and Propagation vol 13 no 6 pp 987ndash989 1965

[15] A K Agrawal H J Price and S H Gurbaxani ldquoTransientresponse of multiconductor transmission lines excited by anonuniform electromagnetic fieldrdquo IEEE Transactions on Elec-tromagnetic Compatibility vol 22 no 2 pp 119ndash129 1980

[16] F Rachidi ldquoFormulation of the field-to-transmission line cou-pling equations in terms of magnetic excitation fieldrdquo IEEETransactions on Electromagnetic Compatibility vol 35 no 3 pp404ndash407 1993

[17] F Rachidi and S V Tkachenko Electromagnetic Field Inter-action with Transmission Lines From Classical Theory to HFRadiation Effects vol 1 WIT Press 2008

[18] C R Paul Analysis of Multiconductor Transmission Lines JohnWiley and Sons New Jersey NJ USA 2nd edition 2008

[19] W Sweldens and R Piessens ldquoWavelet sampling techniquesrdquoin Proceedings of the Statistical Computing Section pp 20ndash29American Statistical Association 1993

[20] I Daubechies Ten Lectures on Wavelets SIAM Press Philadel-phia Pa USA 1992

[21] X Zhu T Dogaru and L Carin ldquoAnalysis of the CDF biorthog-onal MRTD method with application to PEC targetsrdquo IEEETransactions on Microwave Theory and Techniques vol 51 no9 pp 2015ndash2022 2003

[22] A Taflove Computational Electrodynamics The Finite-Difference Time-Domain Method Artech House Norwood NJUSA 1995

[23] A Alighanbari and C D Sarris ldquoRigorous and efficient time-domain modeling of electromagnetic wave propagation andfading statistics in indoor wireless channelsrdquo IEEE Transactionson Antennas and Propagation vol 55 no 8 pp 2373ndash2381 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the resistive terminations the iterative equations for theterminal voltages are derived a method is proposed toupdate the iterative equations which contain some termswhose indices exceed the index range in the MRTD schemeand then the stability and the numerical dispersion arestudied In Section 3 the MRTD scheme is extended tothe two-conductor lossy transmission line In Section 4 thenumerical results are presented on the terminal response ofboth lossless and lossy transmission lines using the MRTDscheme and compared to the FDTDmethod at different spacediscretization numbers and different Courant numbers

2 MRTD Scheme for Two-Conductor LosslessTransmission Lines

21 MRTD Formulation In this section the MRTD schemeis applied to the following scalar transmission lines equationsfor two-conductor lossless lines [18]

120597119881 (119911 119905)

120597119911+ 119897

120597119868 (119911 119905)

120597119905= 0 (1a)

120597119868 (119911 119905)

120597119911+ 119888

120597119881 (119911 119905)

120597119905= 0 (1b)

where 119897 and 119888 are the per-unit-length inductance and capaci-tance respectively

Based on the method outlined in [1] the voltage andcurrent can be expanded as follows

119881 (119911 119905) =

+infin

sum


119881119899

119896120601119896(119911) ℎ119899(119905) (2a)

119868 (119911 119905) =

+infin

sum


119868119899+12

119896+12120601119896+12

(119911) ℎ119899+12

(119905) (2b)

where 119881119899

119896and 119868

119899+12

119896+12are the coefficients for the voltages

and currents in terms of scaling functions respectively Theindices 119899 and 119896 are the discrete spatial and temporal indicesrelated to space and time coordinates via 119911 = 119896Δ119911 and119905 = 119899Δ119905 where Δ119911 and Δ119905 represent the spatial and temporaldiscretization intervals in 119911 and 119905 direction The functionℎ119899(119905) is defined as

ℎ119899(119905) = ℎ (

119905

Δ119905minus 119899) (3)

with the rectangular pulse function

ℎ (119905) =

1 for |119905| lt1

21

2for |119905| =

1

2

0 for |119905| gt1

2

(4)

The function 120601119896(119911) is defined as

120601119896(119911) = 120601 (

119911

Δ119911minus 119896) (5)

minus04

minus02

0

02

04

06120601(z)

08

1

12

14

05 1 15 2 25 30z

Figure 1 Daubechiesrsquo scaling function with two vanishingmoments

where 120601(119911) represents Daubechiesrsquo scaling function Figure 1shows Daubechiesrsquo scaling function with two vanishingmoments

For deriving theMRTD scheme for (1a) and (1b) we needthe following integrals

int

+infin

minusinfin

ℎ119899(119905) ℎ1198991015840 (119905) 119889119905 = 120575

1198991198991015840Δ119905

int

+infin

minusinfin

ℎ119899(119905)

120597ℎ1198991015840+12

(119905)

120597119905119889119905 = 120575

1198991198991015840 minus 1205751198991198991015840+1

(6)

where 1205751198991198991015840 represents the Kronecker symbol Consider

int

+infin

minusinfin

120601119896(119911) 1206011198961015840 (119911) 119889119911 = 120575

1198961198961015840Δ119911 (7)

int

+infin

minusinfin

120601119899(119911)

1205971206011198991015840+12

(119911)

120597119911119889119911 =

119871119878minus1

sum

119894=minus119871119878

119886 (119894) 120575119899+1198941198991015840 (8)

where 119871119878denotes the effective support size of the basis func-

tions The coefficients 119886(119894) are called connection coefficientsand can be calculated by (9) Taking Daubechiesrsquo scalingfunctions as the basis functions Table 1 shows 119886(119894) for 0 le

119894 le 119871119878minus 1 which are zeros for 119894 gt 119871

119878minus 1 and for 119894 lt 0 it can

be obtained by the symmetry relation 119886(minus1 minus 119894) = minus119886(119894)

119886 (119894) =1

120587int

infin

0

10038161003816100381610038161003816 (120582)

10038161003816100381610038161003816

2

120582 sin 120582 (119894 +1

2) 119889120582 (9)

where (120582) represents the Fourier transform of 120601(119911)Daubechiesrsquo scaling functions satisfy the shifted interpo-

lation property [19]

120601 (119894 + 1198721) = 1205751198940

(10)

for 119894 integer where1198721= int+infin

minusinfin119911120601(119911)119889119911 is the first moment of

the scaling functions and the values of1198721are listed in Table 1

Following the theory in [3] and making use of (10) (5) ismodified to

120601119896(119911) = 120601 (

119911

Δ119911minus 119896 + 119872

1) (11)



119894 1198632

1198633

1198634


06339743121 08174005815 10053923835119902max 07500 06844 06585


119881 (119896Δ119911 119899Δ119905)

= ∬

+infin

minusinfin

119881 (119911 119905) 120575 (119911

Δ119911minus 119896) 120575 (

119905

Δ119905minus 119899) 119889119911 119889119905

= 119881119899

119896

(12)


119899

119896and 119868119899+12





119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (13a)

119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (13b)



119878and 119877



119899

0 119881119899

1 119881

119899


119899+12

12 119868119899+12

32 119868

119899+12


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

z

VS(t)

RS





119878(119905)




equiv 119881119878(119899Δ119905) and 119868

119899

119878equiv 119868119878(119899Δ119905)


119871equiv 119881119871(119899Δ119905) and 119868

119899



119881119899

0= 119881119899

119878minus 119877119878119868119899

119878(14a)

119881119899

NDZ = 119881119899

119871+ 119877119871119868119899

119871 (14b)



0and

119881119899+1


119899+1

119894and 119881

119899


the currents 119868119899+12

119894+12and 119868

119899+12




119871119878minus1

sum

119894=0

(2119894 + 1) 119886 (119894) = 1 (15)


sum

119894=0

119886 (119894) (2119894 + 1)119881119899+1

119896=

119871119878minus1

sum

119894=0

119886 (119894) (2119894 + 1)119881119899

119896

minus

119871119878minus1

sum

119894=0

Δ119905

(2119894 + 1) Δ119911119888minus1

[119886 (119894) (2119894 + 1)

sdot (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12)]

(16)


Iminus12 V0 I12 V1

minusΔz

2

Δz

2

+

minus

+

minus

z = 0 Δz


IL



12)Δz

IS



119886 (119894) (2119894 + 1)119881119899+1

119896

= 119886 (119894) (2119894 + 1) 119881119899

119896

minus 119886 (119894) (2119894 + 1)Δ119905

(2119894 + 1) Δ119911119888minus1

(119868119899+12

119896+119894+12minus 119868119899+12


(17)

for 119894 = 0 1 119871119878




Taking 119881119899+1



119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119894+12minus 119868119899+12



119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

12minus 119868119899+12

minus12) (19a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

32minus 119868119899+12

minus32) (19b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

minus119871119878+12

)

(19c)



minus12 119868119899+12

minus32 119868

119899+12

minus119871119878+12


119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ1199112119888minus1

(119868119899+12

12minus 119868119899+12

119878) (20a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

(3Δ119911) 2119888minus1

(119868119899+12

32minus 119868119899+12

119878) (20b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ1199112

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

119878)

(20c)


= (119868119899

119878+ 119868119899+1

119878)2 and 119868

119899

119878can


119868119899

119878=

(119881119899

119878minus 119881119899

0)

119877119878

(21)


119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

119894+12minus 119868119899+12

119878) (22)



119881119899+1

0= (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(Δ119911

Δ119905119888119877119878minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0minus 2119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12

+

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(23)


119881119899+1

NDZ = (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(Δ119911

Δ119905119888119877119871minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 2119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12 +

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(24)


Taking 119881119899+1

119896as an example for 119896 = 1 2 119871

119878minus 1

decomposing (13a)

119886 (0) 119881119899+1

119896= 119886 (0) 119881

119899

119896minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

119896+12minus 119868119899+12

119896minus12) (25a)

3119886 (1) 119881119899+1

119896= 3119886 (1) 119881

119899

119896minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

119896+32minus 119868119899+12

119896minus32) (25b)

(2119896 minus 1) 119886 (119896 minus 1)119881119899+1


119899

119896minus1


(2119896 minus 1) Δ119911119888minus1

(119868119899+12

2119896minus12minus 119868119899+12

12)

(25c)

(2119896 + 1) 119886 (119896) 119881119899+1

119896= (2119896 + 1) 119886 (119896) 119881

119899

119896minus (2119896 + 1) 119886 (119896)

sdotΔ119905

(2119896 + 1) Δ119911119888minus1

(119868119899+12

2119896+12minus 119868119899+12

minus12)

(25d)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

119871119878minus1

= (2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899

119871119878minus1

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119896+119871119878minus32

minus 119868119899+12

119896minus119871119878+12

)

(25e)



119878minus 119896


119878minus 119896


119896by using the


119878terms


119881119899+1

119896= 119881119899

119896minus (

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(26)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 119881119899

119896minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(27)


119878+ 2 NDZ minus 1


119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (28)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878



119868119899+12

119896+12= 119868119899minus12

119896+12minus (

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(29)

for 119896 = 0 1 119871119878minus 2


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

LL

z

VS(t)

RS



119868119899+12

119896+12= 119868119899minus12

119896+12minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(30)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (31)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119871and 119871

119871 respectively



119878(119905)



written as follows

1198810(119905) = 119881

119878(119905) minus 119877

119878119868119878(119905) (32a)

119881NDZ (119905) = 119881119871(119905) + 119877

119871119868119871(119905) + 119871

119871

119889119868119871(119905)

119889119905 (32b)

Expanding 1198810(119905) 1198680(119905) 119881119871(119905) and 119868

119871(119905) as (2a) and (2b)


119881119899+1

0+ 119881119899

0

2=

119881119899+1

119878+ 119881119899

119878

2minus 119877119878119868119899+12

119878

(33a)

119881119899+1

NDZ + 119881119899

NDZ2

=119881119899+1

119871+ 119881119899

119871

2+ 119877119871119868119899+12

119871

+119871119871

2Δ119905(119868119899+32

119871minus 119868119899minus12

119871)

(33b)





119868119899+32

119871= 119868119899minus12

119871+

Δ119905

119871119871

((119881119899+1

NDZ + 119881119899

NDZ) minus (119881119899+1

119871+ 119881119899

119871)

minus 2119877119871119868119899+12

119871)

(34)



119881119899+1

NDZ = 119881119899

NDZ

minusΔ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(35)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ) minus (119881119899

119871+ 119881119899minus1

119871)

minus 2119877119871119868119899minus12

119871)

(36)

for 119899 = 2 3 NDT


119881119899+1

119896minus 119881119899

119896

Δ119905= minus

1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (37a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= minus

1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (37b)


119881119899+1

119896minus 119881119899

119896

Δ119905= 120582119881119899+12

119896

(38a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12

(38b)


|Im (120582)| le2

Δ119905 (39)


2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)



119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)


119870+12repre-


and 1198681199110



1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)


|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)



VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)


119902 =VΔ119905

Δ119911(44)



1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)



119897= 120582REALΔ119911



1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)







120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 1198632

1198633

1198634


06339743121 08174005815 10053923835119902max 07500 06844 06585


119881 (119896Δ119911 119899Δ119905)

= ∬

+infin

minusinfin

119881 (119911 119905) 120575 (119911

Δ119911minus 119896) 120575 (

119905

Δ119905minus 119899) 119889119911 119889119905

= 119881119899

119896

(12)


119899

119896and 119868119899+12





119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (13a)

119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (13b)



119878and 119877



119899

0 119881119899

1 119881

119899


119899+12

12 119868119899+12

32 119868

119899+12


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

z

VS(t)

RS





119878(119905)




equiv 119881119878(119899Δ119905) and 119868

119899

119878equiv 119868119878(119899Δ119905)


119871equiv 119881119871(119899Δ119905) and 119868

119899



119881119899

0= 119881119899

119878minus 119877119878119868119899

119878(14a)

119881119899

NDZ = 119881119899

119871+ 119877119871119868119899

119871 (14b)



0and

119881119899+1


119899+1

119894and 119881

119899


the currents 119868119899+12

119894+12and 119868

119899+12




119871119878minus1

sum

119894=0

(2119894 + 1) 119886 (119894) = 1 (15)


sum

119894=0

119886 (119894) (2119894 + 1)119881119899+1

119896=

119871119878minus1

sum

119894=0

119886 (119894) (2119894 + 1)119881119899

119896

minus

119871119878minus1

sum

119894=0

Δ119905

(2119894 + 1) Δ119911119888minus1

[119886 (119894) (2119894 + 1)

sdot (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12)]

(16)


Iminus12 V0 I12 V1

minusΔz

2

Δz

2

+

minus

+

minus

z = 0 Δz


IL



12)Δz

IS



119886 (119894) (2119894 + 1)119881119899+1

119896

= 119886 (119894) (2119894 + 1) 119881119899

119896

minus 119886 (119894) (2119894 + 1)Δ119905

(2119894 + 1) Δ119911119888minus1

(119868119899+12

119896+119894+12minus 119868119899+12


(17)

for 119894 = 0 1 119871119878




Taking 119881119899+1



119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119894+12minus 119868119899+12



119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

12minus 119868119899+12

minus12) (19a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

32minus 119868119899+12

minus32) (19b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

minus119871119878+12

)

(19c)



minus12 119868119899+12

minus32 119868

119899+12

minus119871119878+12


119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ1199112119888minus1

(119868119899+12

12minus 119868119899+12

119878) (20a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

(3Δ119911) 2119888minus1

(119868119899+12

32minus 119868119899+12

119878) (20b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ1199112

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

119878)

(20c)


= (119868119899

119878+ 119868119899+1

119878)2 and 119868

119899

119878can


119868119899

119878=

(119881119899

119878minus 119881119899

0)

119877119878

(21)


119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

119894+12minus 119868119899+12

119878) (22)



119881119899+1

0= (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(Δ119911

Δ119905119888119877119878minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0minus 2119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12

+

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(23)


119881119899+1

NDZ = (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(Δ119911

Δ119905119888119877119871minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 2119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12 +

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(24)


Taking 119881119899+1

119896as an example for 119896 = 1 2 119871

119878minus 1

decomposing (13a)

119886 (0) 119881119899+1

119896= 119886 (0) 119881

119899

119896minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

119896+12minus 119868119899+12

119896minus12) (25a)

3119886 (1) 119881119899+1

119896= 3119886 (1) 119881

119899

119896minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

119896+32minus 119868119899+12

119896minus32) (25b)

(2119896 minus 1) 119886 (119896 minus 1)119881119899+1


119899

119896minus1


(2119896 minus 1) Δ119911119888minus1

(119868119899+12

2119896minus12minus 119868119899+12

12)

(25c)

(2119896 + 1) 119886 (119896) 119881119899+1

119896= (2119896 + 1) 119886 (119896) 119881

119899

119896minus (2119896 + 1) 119886 (119896)

sdotΔ119905

(2119896 + 1) Δ119911119888minus1

(119868119899+12

2119896+12minus 119868119899+12

minus12)

(25d)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

119871119878minus1

= (2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899

119871119878minus1

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119896+119871119878minus32

minus 119868119899+12

119896minus119871119878+12

)

(25e)



119878minus 119896


119878minus 119896


119896by using the


119878terms


119881119899+1

119896= 119881119899

119896minus (

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(26)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 119881119899

119896minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(27)


119878+ 2 NDZ minus 1


119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (28)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878



119868119899+12

119896+12= 119868119899minus12

119896+12minus (

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(29)

for 119896 = 0 1 119871119878minus 2


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

LL

z

VS(t)

RS



119868119899+12

119896+12= 119868119899minus12

119896+12minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(30)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (31)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119871and 119871

119871 respectively



119878(119905)



written as follows

1198810(119905) = 119881

119878(119905) minus 119877

119878119868119878(119905) (32a)

119881NDZ (119905) = 119881119871(119905) + 119877

119871119868119871(119905) + 119871

119871

119889119868119871(119905)

119889119905 (32b)

Expanding 1198810(119905) 1198680(119905) 119881119871(119905) and 119868

119871(119905) as (2a) and (2b)


119881119899+1

0+ 119881119899

0

2=

119881119899+1

119878+ 119881119899

119878

2minus 119877119878119868119899+12

119878

(33a)

119881119899+1

NDZ + 119881119899

NDZ2

=119881119899+1

119871+ 119881119899

119871

2+ 119877119871119868119899+12

119871

+119871119871

2Δ119905(119868119899+32

119871minus 119868119899minus12

119871)

(33b)





119868119899+32

119871= 119868119899minus12

119871+

Δ119905

119871119871

((119881119899+1

NDZ + 119881119899

NDZ) minus (119881119899+1

119871+ 119881119899

119871)

minus 2119877119871119868119899+12

119871)

(34)



119881119899+1

NDZ = 119881119899

NDZ

minusΔ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(35)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ) minus (119881119899

119871+ 119881119899minus1

119871)

minus 2119877119871119868119899minus12

119871)

(36)

for 119899 = 2 3 NDT


119881119899+1

119896minus 119881119899

119896

Δ119905= minus

1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (37a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= minus

1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (37b)


119881119899+1

119896minus 119881119899

119896

Δ119905= 120582119881119899+12

119896

(38a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12

(38b)


|Im (120582)| le2

Δ119905 (39)


2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)



119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)


119870+12repre-


and 1198681199110



1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)


|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)



VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)


119902 =VΔ119905

Δ119911(44)



1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)



119897= 120582REALΔ119911



1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)







120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Iminus12 V0 I12 V1

minusΔz

2

Δz

2

+

minus

+

minus

z = 0 Δz


IL



12)Δz

IS



119886 (119894) (2119894 + 1)119881119899+1

119896

= 119886 (119894) (2119894 + 1) 119881119899

119896

minus 119886 (119894) (2119894 + 1)Δ119905

(2119894 + 1) Δ119911119888minus1

(119868119899+12

119896+119894+12minus 119868119899+12


(17)

for 119894 = 0 1 119871119878




Taking 119881119899+1



119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119894+12minus 119868119899+12



119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

12minus 119868119899+12

minus12) (19a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

32minus 119868119899+12

minus32) (19b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

minus119871119878+12

)

(19c)



minus12 119868119899+12

minus32 119868

119899+12

minus119871119878+12


119886 (0) 119881119899+1

0= 119886 (0) 119881

119899

0minus 119886 (0)

Δ119905

Δ1199112119888minus1

(119868119899+12

12minus 119868119899+12

119878) (20a)

3119886 (1) 119881119899+1

0= 3119886 (1) 119881

119899

0minus 3119886 (1)

Δ119905

(3Δ119911) 2119888minus1

(119868119899+12

32minus 119868119899+12

119878) (20b)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

0= (2119871

119878minus 1) 119886 (119871

119878minus 1)119881

119899

0

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ1199112

sdot 119888minus1

(119868119899+12

119871119878minus12

minus 119868119899+12

119878)

(20c)


= (119868119899

119878+ 119868119899+1

119878)2 and 119868

119899

119878can


119868119899

119878=

(119881119899

119878minus 119881119899

0)

119877119878

(21)


119881119899+1

0= 119881119899

0minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

119894+12minus 119868119899+12

119878) (22)



119881119899+1

0= (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(Δ119911

Δ119905119888119877119878minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0minus 2119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12

+

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(23)


119881119899+1

NDZ = (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(Δ119911

Δ119905119888119877119871minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 2119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12 +

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(24)


Taking 119881119899+1

119896as an example for 119896 = 1 2 119871

119878minus 1

decomposing (13a)

119886 (0) 119881119899+1

119896= 119886 (0) 119881

119899

119896minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

119896+12minus 119868119899+12

119896minus12) (25a)

3119886 (1) 119881119899+1

119896= 3119886 (1) 119881

119899

119896minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

119896+32minus 119868119899+12

119896minus32) (25b)

(2119896 minus 1) 119886 (119896 minus 1)119881119899+1


119899

119896minus1


(2119896 minus 1) Δ119911119888minus1

(119868119899+12

2119896minus12minus 119868119899+12

12)

(25c)

(2119896 + 1) 119886 (119896) 119881119899+1

119896= (2119896 + 1) 119886 (119896) 119881

119899

119896minus (2119896 + 1) 119886 (119896)

sdotΔ119905

(2119896 + 1) Δ119911119888minus1

(119868119899+12

2119896+12minus 119868119899+12

minus12)

(25d)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

119871119878minus1

= (2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899

119871119878minus1

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119896+119871119878minus32

minus 119868119899+12

119896minus119871119878+12

)

(25e)



119878minus 119896


119878minus 119896


119896by using the


119878terms


119881119899+1

119896= 119881119899

119896minus (

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(26)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 119881119899

119896minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(27)


119878+ 2 NDZ minus 1


119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (28)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878



119868119899+12

119896+12= 119868119899minus12

119896+12minus (

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(29)

for 119896 = 0 1 119871119878minus 2


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

LL

z

VS(t)

RS



119868119899+12

119896+12= 119868119899minus12

119896+12minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(30)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (31)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119871and 119871

119871 respectively



119878(119905)



written as follows

1198810(119905) = 119881

119878(119905) minus 119877

119878119868119878(119905) (32a)

119881NDZ (119905) = 119881119871(119905) + 119877

119871119868119871(119905) + 119871

119871

119889119868119871(119905)

119889119905 (32b)

Expanding 1198810(119905) 1198680(119905) 119881119871(119905) and 119868

119871(119905) as (2a) and (2b)


119881119899+1

0+ 119881119899

0

2=

119881119899+1

119878+ 119881119899

119878

2minus 119877119878119868119899+12

119878

(33a)

119881119899+1

NDZ + 119881119899

NDZ2

=119881119899+1

119871+ 119881119899

119871

2+ 119877119871119868119899+12

119871

+119871119871

2Δ119905(119868119899+32

119871minus 119868119899minus12

119871)

(33b)





119868119899+32

119871= 119868119899minus12

119871+

Δ119905

119871119871

((119881119899+1

NDZ + 119881119899

NDZ) minus (119881119899+1

119871+ 119881119899

119871)

minus 2119877119871119868119899+12

119871)

(34)



119881119899+1

NDZ = 119881119899

NDZ

minusΔ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(35)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ) minus (119881119899

119871+ 119881119899minus1

119871)

minus 2119877119871119868119899minus12

119871)

(36)

for 119899 = 2 3 NDT


119881119899+1

119896minus 119881119899

119896

Δ119905= minus

1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (37a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= minus

1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (37b)


119881119899+1

119896minus 119881119899

119896

Δ119905= 120582119881119899+12

119896

(38a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12

(38b)


|Im (120582)| le2

Δ119905 (39)


2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)



119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)


119870+12repre-


and 1198681199110



1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)


|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)



VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)


119902 =VΔ119905

Δ119911(44)



1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)



119897= 120582REALΔ119911



1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)







120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881119899+1

0= (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(Δ119911

Δ119905119888119877119878minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0minus 2119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12

+

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(23)


119881119899+1

NDZ = (

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(Δ119911

Δ119905119888119877119871minus

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 2119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12 +

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(24)


Taking 119881119899+1

119896as an example for 119896 = 1 2 119871

119878minus 1

decomposing (13a)

119886 (0) 119881119899+1

119896= 119886 (0) 119881

119899

119896minus 119886 (0)

Δ119905

Δ119911119888minus1

(119868119899+12

119896+12minus 119868119899+12

119896minus12) (25a)

3119886 (1) 119881119899+1

119896= 3119886 (1) 119881

119899

119896minus 3119886 (1)

Δ119905

3Δ119911119888minus1

(119868119899+12

119896+32minus 119868119899+12

119896minus32) (25b)

(2119896 minus 1) 119886 (119896 minus 1)119881119899+1


119899

119896minus1


(2119896 minus 1) Δ119911119888minus1

(119868119899+12

2119896minus12minus 119868119899+12

12)

(25c)

(2119896 + 1) 119886 (119896) 119881119899+1

119896= (2119896 + 1) 119886 (119896) 119881

119899

119896minus (2119896 + 1) 119886 (119896)

sdotΔ119905

(2119896 + 1) Δ119911119888minus1

(119868119899+12

2119896+12minus 119868119899+12

minus12)

(25d)

(2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899+1

119871119878minus1

= (2119871119878minus 1) 119886 (119871

119878minus 1)119881

119899

119871119878minus1

minus (2119871119878minus 1) 119886 (119871

119878minus 1)

Δ119905

(2119871119878minus 1) Δ119911

sdot 119888minus1

(119868119899+12

119896+119871119878minus32

minus 119868119899+12

119896minus119871119878+12

)

(25e)



119878minus 119896


119878minus 119896


119896by using the


119878terms


119881119899+1

119896= 119881119899

119896minus (

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(26)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 119881119899

119896minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(27)


119878+ 2 NDZ minus 1


119881119899+1

119896= 119881119899

119896minus

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (28)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878



119868119899+12

119896+12= 119868119899minus12

119896+12minus (

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(29)

for 119896 = 0 1 119871119878minus 2


+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

LL

z

VS(t)

RS



119868119899+12

119896+12= 119868119899minus12

119896+12minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(30)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (31)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119871and 119871

119871 respectively



119878(119905)



written as follows

1198810(119905) = 119881

119878(119905) minus 119877

119878119868119878(119905) (32a)

119881NDZ (119905) = 119881119871(119905) + 119877

119871119868119871(119905) + 119871

119871

119889119868119871(119905)

119889119905 (32b)

Expanding 1198810(119905) 1198680(119905) 119881119871(119905) and 119868

119871(119905) as (2a) and (2b)


119881119899+1

0+ 119881119899

0

2=

119881119899+1

119878+ 119881119899

119878

2minus 119877119878119868119899+12

119878

(33a)

119881119899+1

NDZ + 119881119899

NDZ2

=119881119899+1

119871+ 119881119899

119871

2+ 119877119871119868119899+12

119871

+119871119871

2Δ119905(119868119899+32

119871minus 119868119899minus12

119871)

(33b)





119868119899+32

119871= 119868119899minus12

119871+

Δ119905

119871119871

((119881119899+1

NDZ + 119881119899

NDZ) minus (119881119899+1

119871+ 119881119899

119871)

minus 2119877119871119868119899+12

119871)

(34)



119881119899+1

NDZ = 119881119899

NDZ

minusΔ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(35)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ) minus (119881119899

119871+ 119881119899minus1

119871)

minus 2119877119871119868119899minus12

119871)

(36)

for 119899 = 2 3 NDT


119881119899+1

119896minus 119881119899

119896

Δ119905= minus

1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (37a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= minus

1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (37b)


119881119899+1

119896minus 119881119899

119896

Δ119905= 120582119881119899+12

119896

(38a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12

(38b)


|Im (120582)| le2

Δ119905 (39)


2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)



119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)


119870+12repre-


and 1198681199110



1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)


|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)



VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)


119902 =VΔ119905

Δ119911(44)



1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)



119897= 120582REALΔ119911



1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)







120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ + +

minus

+minus

minus minus

I(0 t)

V(0 t)

I(z t)

I(z t)

V(z t)

I(L t)

V(L t)

z = 0 z = L

RL

LL

z

VS(t)

RS



119868119899+12

119896+12= 119868119899minus12

119896+12minus (


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(30)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 119868119899minus12

119896+12minus

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (31)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119871and 119871

119871 respectively



119878(119905)



written as follows

1198810(119905) = 119881

119878(119905) minus 119877

119878119868119878(119905) (32a)

119881NDZ (119905) = 119881119871(119905) + 119877

119871119868119871(119905) + 119871

119871

119889119868119871(119905)

119889119905 (32b)

Expanding 1198810(119905) 1198680(119905) 119881119871(119905) and 119868

119871(119905) as (2a) and (2b)


119881119899+1

0+ 119881119899

0

2=

119881119899+1

119878+ 119881119899

119878

2minus 119877119878119868119899+12

119878

(33a)

119881119899+1

NDZ + 119881119899

NDZ2

=119881119899+1

119871+ 119881119899

119871

2+ 119877119871119868119899+12

119871

+119871119871

2Δ119905(119868119899+32

119871minus 119868119899minus12

119871)

(33b)





119868119899+32

119871= 119868119899minus12

119871+

Δ119905

119871119871

((119881119899+1

NDZ + 119881119899

NDZ) minus (119881119899+1

119871+ 119881119899

119871)

minus 2119877119871119868119899+12

119871)

(34)



119881119899+1

NDZ = 119881119899

NDZ

minusΔ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(35)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ) minus (119881119899

119871+ 119881119899minus1

119871)

minus 2119877119871119868119899minus12

119871)

(36)

for 119899 = 2 3 NDT


119881119899+1

119896minus 119881119899

119896

Δ119905= minus

1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (37a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= minus

1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (37b)


119881119899+1

119896minus 119881119899

119896

Δ119905= 120582119881119899+12

119896

(38a)

119868119899+12

119896+12minus 119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12

(38b)


|Im (120582)| le2

Δ119905 (39)


2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)



119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)


119870+12repre-


and 1198681199110



1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)


|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)



VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)


119902 =VΔ119905

Δ119911(44)



1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)



119897= 120582REALΔ119911



1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)







120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

35

40


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60


Phas

e err

or (d

egre

es)

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)



119881119870

= 1198811199110

exp [minus119895 (119896119911119870Δ119911)] (40a)

119868119870+12

= 1198681199110

exp [minus119895 (119896119911(119870 +

1

2)Δ119911)] (40b)


119870+12repre-


and 1198681199110



1205822

= minus4

119897119888Δ1199112[

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911)]

2

(41)


|Im (120582)| le2VΔ119911

119871119878minus1

sum

119894=0

|119886 (119894)| (42)



VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(43)


119902 =VΔ119905

Δ119911(44)



1

VΔ119905sin(

120596Δ119905

2) =

1

Δ119911

119871119878minus1

sum

119894=0

119886 (119894) sin(119896119911(119894 +

1

2)Δ119911) (45)



119897= 120582REALΔ119911



1

119902sin

120587119902

119899119897

=

119871119878minus1

sum

119894=0

119886 (119894) sin [(2119894 + 1)120587119906

119899119897

] (46)







120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120597119881 (119911 119905)

120597119911+ 119903119868 (119911 119905) + 119897

120597119868 (119911 119905)

120597119905= 0 (47a)

120597119868 (119911 119905)

120597119911+ 119892119881 (119911 119905) + 119888

120597119881 (119911 119905)

120597119905= 0 (47b)



119899

119896

and 119868119899+12



119881119899+1

119896= 1198751119881119899

119896

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(48a)

119868119899+12

119896+12= 1198761119868119899minus12

119896+12

minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(48b)



1198751= (1 +

Δ119905

2119892119888minus1

)

minus1

(1 minusΔ119905

2119892119888minus1

) (49a)

1198752= (1 +

Δ119905

2119892119888minus1

)

minus1

(49b)

1198761= (1 +

Δ119905

2119903119897minus1

)

minus1

(1 minusΔ119905

2119903119897minus1

) (50a)

1198762= (1 +

Δ119905

2119903119897minus1

)

minus1

(50b)


1

1198752and 119876




119878and 119877




119881119899+1

0= (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119878)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119878minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

0

minus 21198752119877119878

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

119894+12+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119878+ 119881119899

119878)]

(51)


119881119899+1

NDZ = (1198752

119871119878minus1

sum

119894=0

119886 (119894) +Δ119911

Δ119905119888119877119871)

minus1

sdot [(1198751

Δ119911

Δ119905119888119877119871minus 1198752

119871119878minus1

sum

119894=0

119886 (119894))119881119899

NDZ

+ 21198752119877119871

119871119878minus1

sum

119894=0

119886 (119894) 119868119899+12

NDZminus119894+12

+ 1198752

119871119878minus1

sum

119894=0

119886 (119894) (119881119899+1

119871+ 119881119899

119871)]

(52)


119881119899+1

119896= 1198751119881119899

119896minus 1198752(

119896minus1

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1

119896minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(53)

for 119896 = 1 2 119871119878minus 1


119881119899+1

119896= 1198751119881119899

119896minus 1198752(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119888minus1


119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(54)


119878+ 2 NDZ minus 1



119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881119899+1

119896= 1198751119881119899

119896minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12

119896minus119894minus12) (55)

for 119896 = 119871119878 119871119878+ 1 NDZ minus 119871

119878


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(

119896

sum

119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1

119896

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(56)

for 119896 = 0 1 119871119878minus 2


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762(


119894=0

(2119894 + 1) 119886 (119894))

minus1

Δ119905

Δ119911

sdot 119897minus1


119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(57)


119878+ 2 NDZ minus 1


119868119899+12

119896+12= 1198761119868119899minus12

119896+12minus 1198762

Δ119905

Δ119911119897minus1

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894) (58)

for 119896 = 119871119878minus 1 119871

119878 NDZ minus 119871

119878


119881119899+1

NDZ = 1198751119881119899

NDZ

minus 1198752

Δ119905

Δ119911119888minus1

119871119878minus1

sum

119894=0

2119886 (119894) (119868119899+12

NDZminus119894+12 minus 119868119899+12

119871)

(59)

where

119868119899+12

119871= 119868119899minus32

119871+

Δ119905

119871119871

((119881119899

NDZ + 119881119899minus1

NDZ)

minus (119881119899

119871+ 119881119899minus1

119871) minus 2119877

119871119868119899minus12

119871)

(60)

for 119899 = 2 3 NDT


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905

= minus1

119888Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119868119899+12

119896+119894+12minus 119868119899+12


(61a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905

= minus1

119897Δ119911

119871119878minus1

sum

119894=0

119886 (119894) (119881119899

119896+119894+1minus 119881119899

119896minus119894)

(61b)


119875minus1

2119881119899+1


21198751119881119899

119896

Δ119905= 120582119881119899+12

119896

(62a)

119876minus1

2119868119899+12

119896+12minus 119876minus1

21198761119868119899minus12

119896+12

Δ119905= 120582119868119899

119896+12 (62b)


|Im (120582)| le2

Δ119905 (63)


VΔ119905

Δ119911le

1

sum119871119878minus1

119894=0|119886 (119894)|

(64)


minus1

2= 119875minus1

21198751and 119876

minus1

2= 119876minus1

21198761 that








t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










t

A

120591f120591r

VS(t)


119891

4 Numerical Result


119862= radic119897119888 = 50Ω



119903= 1 120583s and








2

1198633 and 119863






120598 =sum

NDT119894=1

(119909 (119894) minus (119894))2

sumNDT119894=1

2

(119894) (65)


minus5

0

5

10

15

20

25

Volta

ge (V

)

FDTDD2-MRTD

D3-MRTDD4-MRTD

5 10 15 200t (120583s)




1198633-MRTD 066 00054

1198634-MRTD 062 00058




3




2-MRTD increasewith


4-MRTD the


3-MRTD and 119863

4-MRTD



20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

20 30 40 50 60 70 80 90 1000

05

1

15

2

25


Rela

tive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)


02

04

06

08

1

12

14

16

18

2

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

0

05

1

15

2

25

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(b)






4-





4-MRTD




14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










14

16

18

2

22

24

26

28

3

32

34


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD

(a)

1

15

2

25

3

35


Relat

ive e

rror

()

FDTDD2-MRTD

D3-MRTDD4-MRTD


+ +

minus

+minus

minus

V(L t)V(0 t)

20 cm

Zc = 95566ΩRL = 50Ω

LL = 05 120583HVS(t)

= 50ΩRS

(a)

100120583m

10120583m20120583m20120583m20120583m

120576r = 12





3wavelet and 119863





119878=


= 50Ω



t05nsO

1V

VS(t)



119879119863



=








0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

14So

urce

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)

(a)

minus02

0

02

04

06

08

1

12

Load

vol

tage

(V)

FDTDD3-MRTDD4-MRTD

5 10 15 200t (120583s)




(NDZ = 40 119902 = 05)

Analysisscheme



Runtime (s)

FDTD 030 092 000661198633-MRTD 018 057 00245

1198634-MRTD 017 054 00275








3-MRTD and119863

4-MRTD


3-MRTD and



4



5 Conclusion



005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










005

01

015

02

025

03


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1


Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(b)


005

01

015

02

025

03

035

04

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD

(a)

02

03

04

05

06

07

08

09

1

0101502025030350404505Courant number

Relat

ive e

rror

()

FDTDD3-MRTDD4-MRTD


Competing Interests


Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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