616 WSEA3 TRANSACTIONS ON MATHEMATICS Issue 3, Vol. 3, July 2004 ISSN: 1109·2769

MORn: AGOS SALIM NASIR}, AHMAD IZANI MD. ISMAIL2

IDepartment ofEngineering Sciences,Universiti Teknologi MARA,

Penang Branch Campus, 13500 Pennatang Pauh, Pulau Pinang,·MALAYSIA

2School ofMathematicai Sciences,Universiti Sains Malaysi~ 11800 Minden, Pulau Pinang,

MALAYSIA

.mohdagossalimnasir@yahoo.com, izani@cs.usm.my

Abstract:-The Goursat problem, associated with hyperbolic partial differential equations, arises in several areas..a( applications. These include mathematical modeling of reacting gas flows and supersonic flow. Recent~erical studies of the problem have focused on the implementation and accuracy aspects of various finitedifference· schemes. ~eoretical considerations such as· ~abi1ity, consistency and convergence have notreceived much attention. In this paper we consider the theoretical aspects.ofa numerical scheme widely used tosolve the problem by considering its application to model linear problem . We obtain results relating to thestability (using Von Neumann stability analysis), consistency and convergence of the scheme. We verify thesetheoretical results with data from computational experiments.

Key- Words:- Finite difference schemes; partial differential equations; means; stability;Consistency; convergence

1 Introduction

The Goursat problem, associated with hyperbolicpartial differential equations, arises· in several areas ofphysics and engineering. An and !Ida (1981) and Chenand Li (2000) describes in detail how this equationarises and is used in the mathematical modeling of&ting gas flows and supersonic flows respectively.~ investigate these mathematical· models in greaterdetail, we need to resort to numerical techniques and inthis regard the finite difference method is often used.The standard· finite difference scheme for the Goursatproblem is a method .based on arithmetic averaging offunction values and a recent study (Ismail et. aI., 2004)has reaffinned the advantages of this method (" theAM scheme").

What a~surance is there that the solution obtained by anumerical method is close to the exact solution? Oneway would be to check that the computed results

converge when the grid sizes are reduced. However..clearly it would .be better if convergence can beguaranteed beforehand. It is well-lmown in numericalanalysis that for linear .problems this guarantee can begiven if the numerical method is stable and consistent.The concept of stability is concerned with the growth, ordecay, of errors (produced because the computer cannotgive answers to an infinite number of decimal places) atany· stage of the computation (Fletcher, 1990). Thesystem of algebraic equations generated by thediscretisation process is said to be consistent with theoriginal partial differential equation if, in the limit that thegrid spacing tends' to zero, the system of algebraicequations is equivalent to the partial differential equarior.at each grid point (Fletcher, 1990). Consistency isconcerned with how well the algebraic equationsapproximates the partial differential equation.

Recent studies of numerical methods for the Goursar

problem ( Ismail et.a!., 2004 ; Wazwaz, 1993 ) ha\'e

WSEAS TRANSACTIONS ON MATHEMATICS Issue 3 Vol. 3, July 2004 JSSNj 1109·2769 617

focused on implementation and accuracy of varioustinite' difference schemes. In this paper we willUlvestigate the stability, consistency and convergenceof. the AM' scheme when applied to a model linearGoursat problem. .

., The Goursat Problem and the.-

AM Scheme

I The Goursat problem is of the form rv.razwaz, 199~):

U lty =f(x,y,u,u,pu y )

u(x,O) = cp(x), u(O, y) =\V(y), 4>(0) =weO)

°S x S a,O ~ y S b... (1)

The established finite difference scheme is based onI arithmetic mean averaging of function values and is

given by (Ismail et.a12004):

±(fi+1,j+l + fi,j + fi+1,j + fi,i... l ) ..

·... (2)

. j denotes the grid size. As was mentioned earlier, weshall refer to the finite difference scheme (2) as the M1scheme. For the AM scheme, .the function value at 'grid:ocation (i + 1/ 2, j +1/ 2) is given by :

The analytical solution of (4) is e HY (Wazwaz, 1995)

the AM scheme for the partial differential equa~on in (4is:

'U i+1,j+J +,Ui,j - Ui+l,j - Ui,j+-l _

h2 -

±(~i+I.;'1 +ui.J +Ui+I.J +Ui•J+1)...(5J

Equation (5) can be rewritten as:

...(6)

l+r ' h 2

where A=-- with r=->O.1-r 4

~ Stability

The stability of a fmite difference scheme can beinvestigated using the Von Neumann method (Fletcher,1990). In this method, the errors distributed along gridlines at one time level are expanded as a finite Fourierseries. If the separate Fourier components of the errordistribution amplify in progressing to the next time level,then the sch~me is unstable.

The error equation for equation (6) is:

where <;i,i is the error at the (iJ) grid point. We write

C; i,j as }} e.He.i where A. is the "amplification factor

for" the rnth Fourier mode of the error distribution as it·propagates one step forward in time and em = m1th. For

linear schemes it is. sufficient to consider the propagationof the error due to just a single term of the Fourier seriesrepresentation i.e. the subscript m can be dropped. ""

... (3)

-\\"e shall investigate the stability consistency and~onvergence of the A...\1 scheme for linear Goursat~roblems by considering the model linear Goursat~roblem: "

Ux.y =u

C;i...l.j...l =A(Si+l,j + C;i,j+r) - <;ij ...(7)

u(x.O) =e''(

u(O. y) =e Y

o::; x S 2. 0 ~ y::; 2... (4)

Substituting <;i.j =')e.j:j8i into equation (7) gives;

618 WSEAS TR.A.NSACTJONS ON MATHEMATlCS Issue 3, Vol. 3 July 2004 ISSN: 1109·2769

.(9)

Substituting the exact solution into (5) leads to:

[note:.- all terms involving u in equation (10) areevaluated at (Xi' Yj)]

5 Convergence

.- .(101.

As h~ 0, equation (10) becomes U x..j = u. Thus _the

condition for consistency is satisfied.

A solutio~ of the algebraic equations which approximate. a' given partial differential equation is said to be.

convergent if the approximate solution approaches theexact solution for each value of the independent variablesas the .grid spacing tends to zero. The Lax EquivalenceTheorem- (Richtmyer and Morton, 1967) states that given

. a· properly (well) posed linear initial value problem and afinite :difference equation that satisfies the -consistencycondition, stability is the necessary and sufficientcondition for convergence. .

required that

which implies

Ae..H9 -1A=~=--­

e.r::ie - A

...(8)

"J1-~ e ..[.Tl)(iTl) = A ())e.r::i 9(iTl) + A(jTl) e .r::iei )

. ..,) e.J:i 6ii.e.

4 Consistency

To test for consistency) the exact solution of the parti~ldifferential equation is substituted into the fmite

Ofference scheme and values at grid points· expan~edas a Taylor· series. For consistency, the exp~esslOn

obtained should tend to the partial differential equationas the grid sIze tends to zero (Twizell, 1984)..

Squaring both sides and after some manipulation we

obtain that for IAI ~ 1va) A must'satisfy A '1. ;;:; 1 . Since

.•= 1+ r and r > '0) we obtain that the scheme isl-r

stable Vr (except r =1)

For stability . it .isq Ae.f=ie -1AI ~ I'Ve i.e../=iI ::;1"1'6e -19 -A.

Ie.r::ie - AI ~ IAeJ=ie ~ 1\

Expanding as a Taylor series about (xj'Yj) gives:That the Goursat problem -is wen posed was establishedby Garabedian (1964)' by transforrrung it into arintegrodifferential .equation and then solving by .. theteclmique of successive approximations. We: haveestablished that the AM scheme for the linear Goursat

. problem (4) is both stable and consistent. Thus fro~ t~~·Lax .Equivalence Theorem we can conclude It l~

convergent.

WSEAS .TRANSACTIONS ON MATHEMATICS Issue 3. Vol. ). July 2004 ISSN: 1109·2769 619

6 Computational Experiments

A computer program using the AM scheme to solve (4)was developed and the computed results are as follows:

- absolute errors ·at grid pointsh (0.25.0.25) (0.5.0.5) (0.7S.0.7S) ..{I.0.1.0)

0.500 6.923387Se- 6.9233875e- 2.0437S96e- 2.0437S96e-003 003 002 002

0.150 S.lSS7520e- 1.7070573e- 3.24933S4e- 4.9962137e-004 003 003 003

0.\00 5.5301 S66e- 2.7212902e- S.7076620e- 7.9460331 e-005 004 004 004

- v.OSO 2.068406Se- 6.799709ge- 1.2925661 c- 1.9848210e-OOS 005 ·004 004·

-0.025 5.17056Se· 1.6997083e- 3.230.&63Se- 4.9610004e-006 005 005 005-

.abs·olute errors at grid pointsh (1.25.1.25) ·(L5~l.S) (1.75,1.75) (2.0,2.0)

0.500 3.6461034e- 3.6461304e- S.3882494e- 5.3882494c-~ 002 002 002 . 002

·'0.150 6.8729164e- 8.8393041 e- 1.087220ge- 1.2957280e-003 003 002 002

0.100 1.1530440e- 1.402621ge- 1.7883943e- . 2.0515447e-003 003 003 003

0.050 . 2.7267983e- 3.502.446ge- 4.3025 I04e- 5.121255ge-004 . 004 004 004

0.025 6.8 152731e- 8.7535555e- 1.0752707e- 1.2798397e-005 OOS 004 004

Table 1: h values and absolute errors at various gridpoints.

Although tesults at only ,eight points are displayed,numerical experiments indicate that the absolute errorbecomes smaller as· his decreased for all grid pointstested. '

7 Conclusions

aevious studies of the finite difference solution of the,"oursat problem· have focused on the accuracy and

:mplementation aspects. In'this paper we have studiedthe theoretkal aspects of the finite difference solution'

.of a linear Goursat problem .using the Mf scheme.l'sing the Von Neumann method we have sho\\TI that itIS unconditionally stable and we have also sho'Wn it isconsistent. Invoking the Lax Equivalen.ce Theorem wededuce that· the scheme is convergent. Numericalexperiments that we have conducted verify that thesd'.eme is convergent

Acknowledgements

We acknowledge ~e support of an FRGS grant.

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