RD-RI75 441 R NUMERICRL INVESTIGATION OF FINITE - VOLUME TECHNIQUES 1/1-USING THE INVISCI.. (U) AIR FORCE RMAMENT LAO EGLIN RFSFL J S MOUNTS ET AL. OCT 86 RFRTL-TR-96-66
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AFATL-TR-86-66
A Numerical Investigation of Finite -Volume Techniques Using the InviscidBurgers Equation
Jon S Mounts, 1 Lt, USAFyMontgomery C. Hughson, 2Lt, USAFR" Dave M Belk
DTIC!LECTE
NOV 13 1WSAERODYNAMICS BRANCH 0
: I AEROMECHANICS DIVISION
OCTOBER 1986
INTERIM REPORT FOR PERIOD OCTOBER 1985-SEPTEMBER 1986
'APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED
OTIC FILE 60.YAIR FORCE ARMAMENT LABORATORYAir Force Systems Command I United States Air Force lEglin Air Force Base, Florida
"" : 1 ....... " __" ".*"..."-'. , N' 41. , .3-
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4. PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S)
AFATL-TR-86-66
& NAME OF PERFORMING ORGANIZATION lb. OFFICE SYMBOL 7. NAME OF MONITORING ORGANIZATIONAir Force Armament Laboratory (ifappliable) Air Force Armament Laboratory
Aerodynamics Branch AFATL/FXA Aeromechanics Division
6c. ADDRESS (City. State and ZIP Code) 7b. ADDRESS (City. State and ZIP Code)
Eglin AFB, FL 32542-5434 Eglin AFB, FL 32542-5434
Sf NAME OF FUNDING/SPONSORING Sb. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION (if applicable)
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62602F 2567. 03 0811. TITLE (Include Security Chaiflcation)A Numerical Investigation of Finite-Volume Techniques Using the Inviscid Burgers Equation (L
12. PERSONAL AUTHOR(S)Jon S. Mounts, Montgomery C. Hughson, and Dave M. Belk
13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Yr., Mo., Day? 15. PAGE COUNT
Interim FROM Oct 85 TO seo 86 October 1986 481S. SUPPLEMENTARY NOTATION
None
17. COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necenary and identify by block number)
FIELD GROUP SUB. GR. Method01 01 Finite-Volume Techniques
Burgers Equation1S. ABSTRACT (Continue on muwere If neceesary and identify by block number)
A numerical investigation of finite-volume techniques using the inviscid Burgers equationwas conducted. An explicit, second-order, one-sided, or upwind differencing schemedeveloped by Warming and Beam was used. First, the difference scheme was analyzed forconsistency, stability, convergence, phase and dispersion error, and artificial dissipationThen, various combinations of finite-volume and extrapolation techniques were tested andconclusions and recommendations for their use were formulated.
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22. NAME OF RESPONSIBLE INDIVIDUAL 22b. TELEPHONE NUMBER 22c. OFFICE SYMBOL(Include Area Code)(JON S. MOUNTS 904) 882-3124 FXA
DD FORM 1473, 83 APR EDITION OF I JAN 73 IS OBSOLETE. UNCLASSIFIED
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E2.
PREFACE
This report was prepared by Jon S. Mounts, Montgomery C. Hughson, andDave M. Belk of the Computational Fluid Dynamics Section, AerodynamicsBranch, Aeromechanics Division, Air Force Armament Laboratory, Eglin AFB,Florida. The work was performed under work unit 25670308 during the fiscalyear period from 1 October 1985 to 30 September 1986.
This report presents the investigation of finite-volume techniquesemployed in current Euler codes, in use in the Computational Fluid DynamicsSection.
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TABLE OF CONTENTS
Section Title Page
I INTRODUCTION . . . ......... . . . . . . ...... 1
II UPWIND SCHEME. . . . . . . . . . . . . . . . . . .*. . . 2
III FINITE VOLUME APPROACHES ..................... 3
1. Extrapolation Techniques ...... ...... . 3
2. Differencing Techniques ....... . . . . . . . 3
IV NUMERICAL RESULTS ........... .................... 8
V CONCLUSION . . . . ................... 26
Appendices
A. INVISCID BURGERS EQUATION. . . . . . ........ 27
B. ANALYSIS OF NUMERICAL TECHNIQUE. . . . . . . 31
1. Linearization ......................... 31
2. Consistency. . . . . . ...... . . . 32
3. Stability ............................ 33
4. Convergence . . . . . . ........ . . 33
5. Modified Equation ... ....... . ........ 34
6. Phase and Dispersion Error ........... 34
7. Artificial Dissipation .... ............ . 36
References........................ . . . 39
.4%%
m- V
.4.
LIST OF FIGURES
Figure Title Page
1 Finite Volume, Two Point Extrapolation Technique ...... 4
2 Dependent Variable Average (DVA) Differencing Technique . . 6
3 Dual Dependent Variable (DDV) Differencing Technique. . . . 7
4 Burgers Equation Solution- V = 1.50. . . . . . . . . . 9
5 Burgers Equation Solution- V - 2.00 . . . . . . . . . . . 13
6 Burgers Equation Solution .................... ... 17
A-I Characteristic Solution to the Inviscid Burgers Equation. , 30
B-i Relative Phase Shift Error vs. Courant Number (v) . . . . . 37
B-2 Amplification Factor Modules - IGI vs. Courant Number (V) , 38
'N0
4.
Vi
,le
LIST OF SYMBOLS AND ABBREVIATIONS
U. Velocity in the x Direction
F Flux
C Constant Wave Speed
FDE Finite Difference Equation
PDE Partial Differential Equation
n Time Step
i Spatial Step (x Direction)
At Change in Time Step (Incremental)
Ax Change in Spatial Step (Incremental)
V Courant Number
Phase Angle of the Amplification Factor
y Wave Number (kjAx)
IGI Amplification Factor
A0 Initial Amplitude of the Fourier Component
vii(The reverse of this page is blank)
.-. "
SECTION I
INTRODUCTION
Researchers in aerodynamic analysis and design have turned to numericaltechniques, with the advent of the supercomputer (Cyber 205, Cray X-MP, Cray2, etc.), to solve for the fluid flow about weapon/store configurations.Current aerodynamic research at the Air Force Armament Laboratory (AFATL) isaimed at solving the equations that govern fluid flow problems using avariety of approximation techniques. The governing partial differentialequations (PDE) form a nonlinear system which must be solved for the unknownpressures, densities, temperatures, and velocities to yield the aerodynamiccharacteristics for a given weapon/store configuration at specific flightconditions.
To obtain a thorough approximation for the flow field about a configu-ration, we must solve the complete Navier-Stokes equations. However, due tolimitations placed on researchers by current computer systems (time andstorage), certain simplifying assumptions must be made to obtain results.By assuming an inviscid, adiabatic flow field (dropping viscous and heattransfer terms), we obtain the Euler equations. Results obtained from asolution of the Euler equations are particularly useful in preliminarydesign work where information on pressure alone is desired. These equationsare also of interest because they incorporate many major fluid dynamics ele-ments such as internal discontinuities (shock waves and contact surfaces).The Euler equations govern the motion of inviscid, non-heated gas and havedifferent numerical characteristics in different flow regimes. For steadyproblems, the equations are elliptic in subsonic flow and hyperbolic insupersonic flow (Reference 1).
Research at AFATL is aimed at solving the three dimensional Eulerequations to approximate the flow about arbitrarily-shaped weapon/storeconfigurations. This research is accomplished using various implicit andexplicit finite-difference and finite-volume techniques. Currently thiswork has lead to an analysis of two explicit, finite-volume procedures forsolving the Euler equations. To help in this analysis of the numericalcharacteristics inherent to the two approaches, a single equation servingas a numerical analog to the Euler equations has been found. The inviscidBurgers equation (nonlinear wave equation) serves as this simple nonlinearanalog to aid in our understanding of these techniques (Appendix A)(Reference 1).
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SECTION II
UPWIND SCHEME
Several numerical techniques have been developed that will solve thepartial differential equations that govern fluid flow, heat transfer, andcombustion problems. These methods can be either explicit or implicit,central difference or upwind, single or multi-step, and first- or second-order accurate.
An explicit, second-order, one-sided, or upwind difference scheme forthe numerical solution of hyperbolic systems has been developed by Warmingand Beam (Reference 2). There are several advantages to the use of theUpwind schemes.
(1) One-sided schemes are often desirable along both fixed externalboundaries and along moving internal boundaries (such as shocks), where aspatially centered scheme would require one or more points inside or acrossthe boundary.
(2) An explicit, second-order upwind scheme can have twice thestability bound of a symmetric scheme using the same number of spatial gridpoints.
(3) The dissipative-dispersive properties of an upwind scheme aresuperior to those of a symmetric scheme.
(4) switching schemes across a discontinuity can reduce the spuriousoscillations usually associated with a second-order accurate shock-capturingtechnique. Warming and Beam's goal was to develop a hybrid scheme whichexploits the advantages of a second-order upwind scheme for aerodynamicflows. This multi-step procedure applied to the inviscid Burgers equationis shown as follows:
n+l n t nPredictor: U =U - -% F (la)i Ax I
Corrector: (ib)Un U + U At nU.U.F --
I i 2Ax I2
At A2 n+lA F.
where; 2Ax a.
F. F. - Fi (2)
A thorough error analysis has been performed on the Upwind scheme to
aid in our understanding of the numerical characteristics inherent to thistechnique (Appendix B). This analysis includes consistency, convergence,numerical stability, phase and dispersion error, and artificial dissipa-tion (Reference 3).
21 J
SECTION III
FINITE-VOLUME APPROACHES
For our research in computational aerodynamics, we have settled on thefinite-volume (FV) approach, as opposed to the finite-difference (FD)
method, to solve for the physics of the flow about a weapon/store configura-tion. Fundamentally, the primary difference between the two methods is that
for the FV approach we solve for the flux at the face of a cell; the FD
approach solves for the flux at the center of the cell (Figure 1). To
accomplish this FV technique we must extrapolate either the flux or thedependent variable from the center of the cell to the face of the cell. The
advantage to using the FV method is that, inherent to the approach, the
conservative property of the PDE is fully maintained.
1. EXTRAPOLATION TECHNIQUES
As mentioned above, there are two extrapolation techniques that must bestudied to determine which yields the optimum results. The flux term isU(i) 2 /2 where the dependent variable is U(i). For the upwind scheme,extrapolating the flux yields
2 (U(i)2/Z) - (U(i-1) 2/2) or (3)
2 (U(i+1) 2/2) - (U(i+2) 2/2)
depending on the direction of the flow of information. When extrapolatingthe dependent variable for the same inviscid Burgers equation, we get;
(2U(i) - U(i-1)) 2/2 oi. (4)
(2U(i+1) - U(i+2)) 2/2,
again, depending on the direction of the flow information. In this way, thedependent variable extrapolation technique yields
2U(i) 2 -2U(i)U(i-1) + U(i-1) 2/2 or (5)
2i+)2 22U(i+)2 -2U(i+1)U(i+2) + U(i+2) /2
This result shows an extra term, - 2U(i+1)U(i+2) , (as compared to Equation3) which will have some effects that are inherent to this type of numericalmethod.
2. DIFFERENCING TECHNIQUES
The flow solvers (Euler codes), being examined by our research, use twotypes of differencing techniques to solve for the flux at the face of acell; both of these techniques have been developed by Whitfield (References4, 5, and 6).
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The first differencing technique (Figure 2) employs a dependent variableaveraging (DVA) approach to determine the direction of the flow of informa-tion across a cell face. For a specific cell face, the value of the depen-dent variable (U) is averaged from both sides of the cell. This yields avalue for the dependent variable and, depending on the sign of U, is used toextrapolate (using one of the techniques discussed above) from one side ofthe cell face or the other, to obtain a value for the flux at the face ofthe cell.
,* The second differencing technique (Figure 3) employs a dual dependent-variable technique (DDV) in which the value for the dependent variable (U)
is determined for both sides of the cell face and, depending on the sign ofU, can utilize both values of the dependent variable if the direction of theflow of information is toward the cell face from both directions. Thedirection of the flow of information determines whether the value of thedependent variable, U, is used from one side of the cell face (or the other)or from both sides, to extrapolate the flux to the cell face.
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NUMERICAL RESULTS
For this analysis, both the extrapolation and differencing techniques areexamined using the inviscid Burgers equation to model the propagation of awave in time. The numerical analysis used here to study the inviscidBurgers equation should yield second-order accurate results. This level ofaccuracy is typified by dispersion or ringing effects which overshoot theactual, physical results.
The first phase of our study looks at forcing a wave to propagate in onlyone direction. This allows us to better examine the differences between thetwo extrapolation techniques; since the wave is moving in only one direc-tion, the differencing techniques are essentially the same. Figure 4a showsthe effects of using the DVA technique with dependent variable extrapola-tion. The results are atypical, for a second-order accurate solutionmethod, in that no dispersive effects (ringing) are apparent downwind of thewave. Figure 4b gives the results for the DVA technique using flux extra-polation. These results show the characteristic dispersive effects yieldedby a second-order scheme in which the numerical solution overshoots theexact solution on the downwind side of the shock. Figure 4c shows resultsfor the DDV technique with dependent variable extrapolation in which weagain observe what appears to be dissipative characteristics to a second-order scheme. In Figure 4d the results are given for the DDV techniqueusing flux extrapolation. Once again the flux extrapolation approach yieldstypical second-order results with ringing effects. To better study theeffects of the extrapolation approaches, a Courant number of 1.50 was usedsince at a Courant number of either 1.00 or 2.00 the numerical techniqueyields the exact solution (by satisfying the shift condition in EquationB.9) (Reference 1).
The second phase of our examination studies the effects of the two differ-encing approaches (DDV and DVA) by forcing two waves to meet at a cell faceof equal velocities (magnitudes). A Courant number of 2.00 is employed sothat the effects of the two extrapolation techniques are negated (by satis-fying the shift condition in Equation B.9) (Reference). Figure 5a shows theresults for the DVA technique using dependent variable extrapolation whichyields the exact solution to the mathematical model. Similar results wereyielded in Figure 5b for the DVA technique using flux extrapolation. Figure5c yields the results for the DDV technique with dependent variable extrapo-lation and shows the exact solution, as does Figure 5d for the DDV techniqueusing flux extrapolation. For this simple model both differencing tech-niques (DDV and DVA) yield the same results; therefore, a more complicatedtest must be accomplished to better understand the limitations of theapproaches.
For the final phase of this investigation, all aspects of the problem areexamined by forcing two waves to approach each other at unequal velocities(magnitudes). Due to the unequal velocities, U , the effective Courantnumber changes for each direction. Figure 6a shows the effects for the
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DDV technique with dependent variable extrapolation. The results show theatypical (dissipative) solution yielded by the dependent variable extrapola-tion. As the waves meet, the wave with the greater magnitude runs over thelesser wave at an average wave speed. In Figure 6b we obtain the resultsfor the DVA technique using flux extrapolation. This approach yields theexpected second-order results and also shows the greater magnitude waverunning over the lesser wave at an average (or deduced) velocity. Figure 6cshows results, similar to those in Figure 6a, for the DDV techniques usingthe dependent variable extrapolation approach. As in Figure 6a, we observethe dissipative effects on the solution and the propagation of the largerwave at an average speed after the collision. Figure 6d shows the resultsfor the DDV technique with flux extrapolation. As in Figure 6b, theexpected dispersive effects are observed and, again, the greater wave movesat an average velocity after the collision.
A further analysis was performed to attempt to find weaknesses in the dif-ferent approaches. In this case the effective Courant number was doubledthereby pushing the CFL condition towards its limit of 2.00. Figure 6eshows results, similar to those in Figure 6a, for the DVA technique withdependent variable extrapolation; however, the propagation of the wave afterthe collision has twice the velocity. In Figure 6f the results are, again,similar to those in Figure 6b with the exception of the wave propogationbeing at twice the speed. Figure 6g shows the only major discrepancy in ouranalysis. The DDV technique with dependent variable extrapolation showsgood results until the meeting point for the unequal waves. At the colli-sion due to diffusive effects, the effective Courant number goes well abovethe stability limit of 2.00 and the solution diverges. However, when theDDV technique is applied with flux extrapolation (Figure 6h), we againobtain typical results similar to those in Figure 6d.
18
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SFCTION V
CONCLUSION
This investigation has yielded several important conclusions dealing withboth the extrapolation techniques and the differencing approaches.
(1) The dependent variable extrapolation technique, used for both thedual dependent variable (DDV) and the dependent variable averaging (DVA)differencing methods, tends to negate the typical dispersive effects foundin second-order accurate schemes. This is primarily due to the extra termfound when applying the dependent variable extrapolation to the inviscidBurgers equation. When applying the flux extrapolation technique to bothdifferencing methods, the expected dispersive effects are obtained.
(2) No significant differences were appareat when comparing the differ-encing techniques (dual dependent variable and dependent variable averag-ing). Both techniques closely modeled the correct mathematical solution,when using the flux extrapolation approach, throughout most of the investi-gation. The only inconsistency in the study occurred when the DDV differ-encing approach was used with dependent variable extrapolation (Figure 6g).In general, this tends to show that the DVA approach may be a more robustmethod than the DDV approach.
Therefore, the recommended approach to solving the inviscid Burgers equationfor this examination is to apply the dependent variable averaging differ-encing technique with flux extrapolation. This method will also be directlyapplied to the three-dimensional Euler equations for the solution of theinviscid flow field about arbitrarily shaped weapon/store configurations.
26
APPENDIX A
INVISCID BURGERS EQUATION
Burgers equation (1948) can serve as a nonlinear analog of the fluidmechanics equations. This single equation has terms that closely duplicatethe physical properties of the fluid equations, i.e., the model equation hasa convective term, a diffusive or dissipative term, and a time-dependentterm. (Reference 1)
Ut + UUx =Uxx (A.1)Unsteady term Convective term Viscous term
Equation (A.1) is parabolic when the viscous term is included and is a goodmodel for the boundary-layer equations, the parabolized Navier-Stokes (PNS)equations and the complete Navier-Stokes equations. If the viscous term isneglected (inviscid Burgers equation), the remaining equation is composed ofthe unsteady term (Ut) and the nonlinear convection term (UU x). Theresulting hyperbolic equation
Ut + UUx = 0; (A.2a)
in conservation law form
Ut + (U 2 /2) = O, (A.2b)
may be considered a simple analog of the Euler eqautions for the flow of aninviscid fluid. The analogy can be drawn due to the fact that both equa-tions are first-order, hyperbolic, quasi-linear, partial differential equa-tions and both model discontinuities, such as shocks, in the flow field(Reference 1).
Burger equation is a partial differential equation because U=U(x,t) and Utand Ux are both used in the equation. It is a first-order equation becauseonly first derivatives appear. Equations (A.2a) and (A. 2b) are nonlinearbecause the unknown variable, or dependent variable, U, is multiplied byitself or its derivatives. Burgers equation belongs to a special class ofnonlinear equations called quasi-linear equations. A quasi-linear equationis one in which the highest order derivatives appear to the first power, asis most easily seen in Equation (A.2b). The three-dimensional Euler equa-tions of inviscid fluid flow are another example of quasi-linear equations.
To provide a good test case for the finite volume approaches, we need toknow some exact solutions to Burgers equation. First, solve Equation (A.2)with the general initial conditions,
U(x,0) = f(x). (A.3)
The solution to the problem will be of the form U=g(x,t). Now consider athree-dimensional space with xt, and U coordinate axes, and define
0 = U - g(x,t). (A.4)
- 27
,o~ %,
The surface 0 (x,t,U)=O in the three-dimensional space defines the solutionto the problem. In other words, any point on the surface, say x0o, to, Uo isa point of the solution given by U(xo,to)= UO. A vector perpendicular tothe surface is given by grad 0 = (Ox' Oy' Oz ) ' and any vector perpendicularto grad 0 must be tangent to the solution surface. If (xo, to, Uo ) is apoint of the solution, and, for infinitesimal displacements, (xo +dx, to+dt,Uo+du) is also a point of the solution, then
grad *.dx = 0,
or *xdx+ 4tdt+ *Udu = 0. (A. 5)
Using Equation (A.4) in Equation (A.5)
-gx-gt + du = 0. (A.6)
6. From Equation (A.2) the following results when U= g(x,t),
gt + Ugx = 0,
from which we can tell that if we take dx, dt, and du in the followingratios, Equation (A.6) will be satisfied:
dt/1 = dx/U = du/0. (A.7)
If r is taken as a parameter along the solution curve, then Equation (A.7)can be written:
dt/dr = 1 , dx/dr = U, du/dr = 0; (A.8)
with the initial conditions:
t = 0, x = xo , U = f(xo).
Solving these equations gives
t(r) = r , U(r) = constant = f(x0 ) and
dx/dr gives
x(r) = rf(x O ) + constant and x(0) = constant = xO .
Therefore;
x(r) = rf(x o ) + xx(t) = tf(x O ) + x0.
For an example, 'take
f(x) = 1-x , 0 < x <
2 x8 > 1
;, 28• ,'.2 %.. ., '- % . " . . _p'. ..- -, , , .' ',. . .'.%.e.% ,.*. . % - ..P. %'e- ' ,
1.) For xo 1 (Figure A-i);
x(t) = tf(x 0 ) + x= xo
U(r) = f(xo) = 0U(x,t) = 0.
2.) For xo <_ 0 (Figure A-i);
x(t) = t + x0U(r) = f(xo ) = 1U(t+xo0 t) = 1
since xo = x-t when U(x,t) 1.
3.) For 0 < x < 1 (Figure A-i);
X = (x-t)/(1-t)
U(x,t) = (4-X)/(.-t).
This method of characteristics analysis shows the Burgers equation capabil-ity to produce shocks or discontinuities in the flow field. This capabilityallows one to test numerical shock-capturing methods using the inviscidBurgers equation and apply them to the three-dimensional Euler equations forinviscid fluid flow.
929
• .4 % " o % " % "-"- " " "°" " "% "% % " % """"% """ "° °
Y,
S SOCK
U=I
0 x
Figure A-1. Characteristic Solution to the Inviscid Burgers Equation
dd%
...
APPENDIX B
ANALYSIS OF NUMERICAL TECHNIQUE
This section of the analysis examines the numerical characteristics of
the Warming-Beam algorithm for the general finite-difference approach
(References 1 and 2). Finite-difference methods involve approximating the
continuous domain of any problem by a discrete domain (grid) and approxi-
mating the PDE's governing any problem by one or more algebraic or finite
difference equations (FDE's). The total error in the solutions of FDE's is
made up of discretization error and stability error. Stability error is
small for stable FDE's since by definition disturbances and errors cannot
grow. Therefore, discretization error accounts for most of the total error.
The discretization error is made up of dissipation and dispersion error.
For this examination the governing PDE is the inviscid Burgersequation:
Ut + (U2 /2)x = 0 . (B•I)
To determine if a FDE is an algebraic analog of a PDE,and is agoodapproximation of the exact solution of the PDE, we must involve the conceptsof consistency, numerical stability, convergence, phase and dispersion error,and artificial dissipation (Reference 3).
1. LINEARIZATION
For the purpose of linear stability theory, we employ a linearizationmethod to the nonlinear PDE,
Ut + UUx = 0.
The resulting linearized PDE is the convection or linear wave equation:
Ut + CU x = 0, (B.2)
and, for continuity, the linear equation will be used throughout the FDEanalysis.
Locally, the PDE may be approximated by the linear PDE with constantcoefficients even though the PDE is globally nonlinear. This approachyields reasonable results; however, the resulting stability criteria (to bediscussed in more detail in subsection 3) is necessary, but not sufficient.The resulting Warming-Beam upwind scheme for the linear wave equation is asfollows:
n+1 n n nPredictor: + C Ui -U_ 1 = 0 (B.3a)
At Ax
31. ..- . - .. .. . .. . . . ,. . .- , .. . .- - -. ., . .. , . -. , .. . .. "- ->_ .-. ..%
n+1 n+1 n n+1 n+1Corrector: Ui - 1/2 (Uj + O) + C Ui -Ui I
At A x2
n n n-C Ui -2Ui _I+ Ui_ 2' (B.3b)
A x
2. CONSISTENCY
To analyze the consistency of the FDE (Equation B.3a,b) used to modelthe PDE (Equation B.2), we must express UP iU in terms ofUi' and its derivatives by using a Taylor series expansion:
Un + l = n + i t+ At "' tUi2.+-U t-+. + U i 3- + • -•( B .4a )i i 3
n + 2 n i " A 2 " 8 3
U. U. + U. 2At + U.4A + U .B "' (B.4b)i i ] 2! i 3!
n n " A 2 "'A 3
U i _ n U i n U I t + U. 2 . - U At + - -(.4c)! i 3:
The reason for this is that the FDE was derived by applying the FIDE at gridpoint i and time level n. Substituting the expansions into the FDE yields
U t + CUx = O (A x2 x At, At 2 ) (B.5)
therefore, Equation (B.5) is mathematically equivalent to the FIDE as thegrid spacing (A x) and time-step size (A t) approach zero.
An FDE is consistent if for every i and n;
lira FDE =PDE (B.6)
0.A x 0• " At 0
Consistency measures the extent to which an FDE approximates a PDE in some
limiting case. A consistency analysis was performed on the upwind schemeand it was found that the upwind scheme, when applied to the convection
~equation, is unconditionally consistent. This means that the
-32
--L
n nlim (FDE)i = (PDE)i (B.7)At - 0AX -~ 0
n n
regardless of how x and t approach zero. Terms such as A tU i or A xUcan be added to the FDE and the resulting FDE will remain unconditionallyconsistent; therefore, there are an infinite number of unconditionallyconsistent FDE's for a given PDE. Consistency alone, however, will notguarantee the accuracy of the solutions of the FDE's since all computationsare performed by using finite A x and A t.
3. STABILITY
Numerical stability is concerned with how errors propagate as the solu-tion is advanced in a time-like variable, and is a concept applicable onlyto parabolic and hyperbolic PDE's. An FDE is stable if the stability errorapproaches zero or does not grow. A given FDE may yield stable or unstablenumerical solutions depending upon the value of some dimensionless parameter
( = c A t/ A x). The need to obtain stable numerical solutions iscritical to the solution of a given PDE since only stable numerical resultshave a chance of being physically meaningful. Conditionally stable FDE's arethose that yield stable solutions when A x and A t are in a given form ( =
c At/ A x). Unconditionally stable FDE's are those that give stable solutionsfor any A x and A t. Unconditionally unstable FDE's give unstable solutionsfor every Ax and A t. Typically, stability bounds for implicit methods areless restrictive than those for explicit methods
There are many different mathematical methods for analyzing thenumerical stability of FDE's. For this examination, the Von Neumann orFourier method is employed. In the Fourier method, the numerical stabilityof an FDE is analyzed by introducing a disturbance into the numerical solu-tion at every grid point in the spatial domain at some arbitrary time level.The disturbance is expanded into a Fourier series and each Fourier componentof that series is analyzed separately. The FDE is stable if all of theFourier components do not grow in time and unstable if any one of theFourier components grows in time. The method of analyzing each Fouriercomponent separately is valid only when the FDE's are linear with respect tothe dependent variable. For this examination the FDE is not linear and mustbe linearized before the stability property can be determined.
A Fourier stability analysis has been performed on the upwind schemeusing the linear wave equation. Results show that it is conditionallystable with the following conditions:
0 < = c t/A x <2 (B.8)
4. CONVERGENCE
An FDE is convergent if the numerical solution of the FDE approaches theexact solution of the PDE as the time-step size and grid spacing approach
33
zero. A convergent FDE can yield a solution of any desired accuracy byreducing the time-step size (L t) and the grid spacing (A x). The analysisof the convergence of an FDE for a complex PDE is extremely difficult, andconvergence analysis techniques are only available for linear PDE's.
The convergence analysis has been performed for the upwind scheme withrespect to the linear wave equation and results show the FDE to be conver-gent (Reference 1).
If the FDE is well-posed, consistent, and stable (as is the case in thisexamination); the Lax Equivalence Theorem states that the FDE of the linearPDE is convergent. Therefore, according to this theorem, the Warming-Beamupwind scheme should be convergent. For FDE's of quasi-linear and nonlinearPDE's, the Lax Equivalence Theorem serves as an important guideline; hence,consistency and stability are crucial tests for convergence.
5. MODIFIED EQUATION
In the modified equation analysis, a PDE is derived that is mathema-tically equivalent to the FDE to be examined. The resulting PDE is calledthe modified equation.
The modified equation is derived by the following two-step procedure:
(1) Expand each term in the FDE in a Taylor series expansion.
(2) Express all time derivatives (with the exception of the first-order time derivative) in terms of spatial derivatives.
The modified equation for this examination is given by
(x2/6 4/82tU +CU = (c-x /6) (]-,) (2-)) U - ( x /8t) (l-)(2-) U +...(B.9)
t x xxx xxxx
This method of analysis is used exclusively in support of the dissipation
and dispersion error investigations.
6. PHASE AND DISPERSION ERROR
Dispersion is mathematically described by the odd-order spatial deriva-tives. The coefficients of the odd-order spatial derivatives in themodified equation for the FDE must be identical to the corresponding coeffi-cients in the PDE in order for a FDE to have the same dispersive character-istics as the PDE it is to represent. If the corresponding coefficients aredifferent, then the solutions of the FDE's contain dispersion errors. Sincethe coefficients of the odd-order spatial derivatives in the modified equa-tion do not match the corresponding coefficients in the PDE, the FDE con-tains dispersion error. The order of dispersion is equal to the order of
34
• * * -+ ..- .**+ .. +- +., --.. . ',... + .... \. *.. .- . ... . *. . ... . .... . . . -+ +.. . . .. . . .
the coefficients of the lowest odd-order spatial derivative excluding thefirst order spatial derivative. Therefore, the FDE contains second-orderdispersion; the higher the order of dispersion, the lower the dispersionerror.
U + CU = 0t x
(x2/ A4/ 1 2Ut + CU - (cAx /6) (l-v) (2-\) U + (Ax /8At) v( v) (2-v) U + .. 0
The dispersion error is examined by the Von Neumann method (Reference1). The dispersion error for each Fourier component of a disturbance aftern time steps is given by
n(0pde - Ofde) (B.10)
where
Opde = phase angle of the amplification factor (PDE)
Ofde = phase angle of the amplification factor (FDE)
The dispersion error after n time steps, for this analysis, is given by
n - tan 1-2) (+2 (l-v) sin 2 sin 2 (B.11)tan y(1+2 (1-v) sin)2
2The relative phase shift error for a given Fourier component after one
time step is
Ofde / Opde (B.12)
The relative phase shift error for this examination is
tan 1-2 ,(j+2 (l-\) sin 2) sin 2 (B.13)
vsin y (1+2 (1-v) sin2 y
-Y\)The dispersion error is given by these two relationships because the phaseangle of the amplification factor depends only on the odd-order spatialderivatives when the highest time derivative is first-order.
For leading phase error, the relative phase shift error must be greaterthan unity for a given Fourier component (the numerical solution for thatFourier component gives a wave speed greater than the wave speed given bythe exact solution). Lagging phase error results when the relative phaseshift error is less than unity for a given Fourier component (the numericalsolution for that Fourier component gives a wave speed less than the wavespeed given by the exact solution).
The Warming-Beam upwind scheme, with respect to the convection equation,has a lagging phase error when v is greater than one and a predominantly
35
97 7
leading phase error when is less than one. When v does not equal one, therelative phase shift error increases as the wave number, y (Kj Ax),increases (Figure B-I).
7. ARTIFICIAL DISSIPATION
As diffusion spreads a disturbance in every direction, the disturbanceis smoothed out over an increasingly large area. Diffusion reduces spatialgradients and lowers the magnitude of the disturbance by spreading it out;this phenomenon is termed dissipation. Diffusion is mathematically des-cribed by even-order spatial derivatives in a PDE; if the even-order spatialderivatives in the PDE are zero, the PDE will not have any dissipation.
In order for a FDE to have the same dissipative characteristics as thatof the PDE it is modeling, the coefficients of the even-order spatial deriv-atives in the modified equation for the FDE must be identical to the corre-sponding coefficients of the PDE. The PDE for our examination is, again,
Ut + CUx = 0
The modified equation for the upwind scheme is given by (Equationf B.9)
CU= (cx/6) (l-v) (2-\) U + (Ax /8At)v (i-v) (2-v) U -+.Ut +Cx xxx xxxx
Therefore, since the corresponding even-order coefficients are not identical,there is fourth-order dissipation error in the solution of the FDE.
The dissipation error is examined using the Von Neumann method. Thedissipation error for each Fourier component of a disturbance after n timesteps is
Gnpde - Gnfde ) A° , (B.14)
Gpde = amplification factor (PDE)
Gfde = amplification factor (FOE)
A0 = initial amplitude of the Fourier component
The dissipation error is given by this relationship because the modulus ofthe amplification factor depends on the even-order spatial derivatives whenthe highest time-derivative is first-order.
The dissipation error, for this examination, for a given time-step, n,
is given by
1- [ (1-4 ')(l- v)2 (2-) sin 4 1/2 ] n (B.15)
For values of ' less than one, as decreases, the dissipation errorincreases, and for values of v greater than one, as v increases, the dis-sipation error inccreases. Also, for values of v not equal to unity, asthe wave number, y (kj A x ), increases the dissipation error increases(Figure B-2).
36
.e J. % -
00
.14
I'a
U4.
U. >
U.Z
37H
%~' %~I %
-Le
-.01
and and and and*..7
y
1.0 0 1.0
AMPLIFICATION FACTOR: G j
Figure B-2. Amplification Factor Modulus - Gjvs Courant Number (')
38
"-,. . '.f/-' -.2 . .' .'._'',,''..''., . 9.'9-. v:'..'4,,. %.J', , ' i .- . - 4. .. .*...', *.-'.,', . '. • - * .
REFERENCES
1. D.A. Anderson, J.C. Tannehill, and R.H. Pletcher, Computational FluidMechanics and Heat Transfer., McGraw-Hill Book Co., New York, 1984.
2. R.F. Warming and R.M. Beam. "Upwind Second-Order Difference Schemes andApplications in Unsteady Aerodynamic Flows," Proc. AIAA 2nd ComputationalFluid Dynamics Conference, Hartford, Conn., 1975,
3. T. I-P. Shih, Finite-Difference Methods in Computational Fluid Dynamics,j to be published by Prentice-Hall Pub.
4. D.L. Whitfield and J.M. Janus, "Three-Dimensional Unsteady EulerEquations Solution Using Flux Vector Splitting," AIAA-84-1552, Jun 1984.
5. W. Schmidt, A. Jameson, and D.L. Whitfield, "Finite Volume Solution forthe Euler Equation for Transonic Flow over Airfoils and Wings IncludingViscous Effects," AIAA-81-1265, Ca., Jun 1981.
6. D.M. Belk, J.M. Janus, and D.L. Whitfield, "Three-Dimensional UnsteadyEuler Equations Solutions on Dynamic Grids," AIAA-85-1704, Apr 86.
39
- -*-*,t .. *... * .. .. . ..
-~ ~~~~ ~~~~~~~ - .- -- . - - . -- -. . . . . . - -
INITIAL DISTRIBUTION
DT IC-DDAC 2
AUL/LSE 1
FTD/SDNF 1
HQ USAFE/INATW 1AFWAL/FIES/SURVIAC 1
*AFATL/DOIL 2
AFATL/CC 1
AFC SA/ SAMI 1AFATL/CCN 1AFATL/FXA 10
*AFATL/FX 2
USAFA/DFAN 1
USAFA/DFAN (PERSONNEL) 1
AD/YHP 1
AFATL/MN 1AD/TYD 1
0'AFOSR/NA 1
NASA AMES RT 1
40
A16'
'dI2%w
V..'*.
