A Survey of the Implementation of Numerical Schemes for Linear Advection Equation

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1 Advances in Pre Mathematics, 4, 4, Pblished Online Agst 4 in SciRes. A Srvey of the Implementation of Nmerical Schemes for Linear Advection Eqation Pedro Pablo Cárdenas Alzate Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia ppablo@tp.ed.co Received 6 Jne 4; revised 6 Jly 4; accepted 3 Agst 4 Copyright 4 by athor and Scientific Research Pblishing Inc. This work is licensed nder the Creative Commons Attribtion International License (CC BY. Abstract The interpolation method in a semi-lagrangian scheme is decisive to its performance. Given the nmber of grid points one is considering to se for the interpolation, it does not necessarily follow that maximm formal accracy shold give the best reslts. For the advection eqation, the driving force of this method is the method of the characteristics, which acconts for the flow of information in the model eqation. This leads natrally to an interpolation problem since the foot point is not in general located on a grid point. We se another interpolation scheme that will allow achieving the high order for the box initial condition. Keywords Nmerical Schemes, Advection Eqation, Semi-Lagrangian Approach. Introdction The classical D linear advection eqation is given by + α = ( t x We can see that Eqation ( is a -D version (linear of the partial differential eqations [], which describe advection of qantities sch as energy, mass, heat, etc. Here, = ( xt,, x and α is a nonzero constant velocity. We can say that Eqation ( describes the motion of a scalar as it is advected by a known velocity field. We know that the niqe soltion of Eqation ( is determined by an initial condition : = ( x, ( ( where xt, = x αt with an arbitrary fnction on. How to cite this paper: Alzate, P.P.C. (4 A Srvey of the Implementation of Nmerical Schemes for Linear Advection Eqation. Advances in Pre Mathematics, 4,

2 An analysis of a nmerical scheme implies to stdy the consistency, stability and accracy. The goal of the analysis is to get an idea abot how good a difference scheme is representing the linear advection eqation. In the following we will make some analysis of the approximate difference schemes to the Eqation (. Nmerical problems arising with polltions models are discssed in []. Textbooks that deal with advection problems are [3] and [4] The spectral-methods and finite elements for linear advection eqations we refer are [5] and [6] respectively. There exists more recent bibliography on finite element methods, which can be conslted in [7]. The paper is organized as follows. In Sections and 3, we consider two cases firstly = cos( x and here we stdy the implementation of the schemes Lax-Friedrich, Lax-Wendroff and RK3-TVD-WENO5 and we derive the expected GTE (Global Trncation Error and we verify it by prodcing a convergence plot (by sing the l, l and l norm. After this, we repeat by sing the box fnction π, x < 4 π π =, x< 4 π, x In Section 4, we consider linear and qadratic interpolation and we stdy the stability and accracy by sing semi-lagrangian approach for linear advection eqation and finally we prodce a convergence plot for ( with another interpolation scheme.. Implementation of the Schemes and GTE Initially we consider the -D linear advection eqation ( ] t x, x S, t, T + = (3 (, ( x = x (4 We can to verify that the soltion is ( xt, = ( x t, and therefore for T ( x, ( x = = [8]. Now, we compte the global trncation error for the schemes as follows. Lax-Friedrich. Here, the discretization of the advection eqation by sing Lax-Friedrich gives and therefore we can verify so the stability condition is finally ( + n+ n n n n j j + j j+ j = xt t x xt x xt ξ (, t = + (, + ( ( +, + (, ( x+t, ( x t, ( ( = O t+ x (7 L-F t = K x ( ( ( ( ( GTE ξ, t = O t+ x = O x (9 Lax-Wendroff. Here, the discretization of the advection eqation by sing Lax-Wendroff gives + n+ n n n n n n j j j+ j j+ j j = + ( ( (5 (6 (8 ( 468

3 and therefore we can verify and the stability condition (CFL is Therefore, (, + (, ( +, (, xt t xt x xt x xt ξ (, t = + L-W ( x+t, ( xt, + ( x t, ( (( (. = O t = K x ( ( (( ( ( ( ( ( GTE ξ, t = O x + x = O x (3 3 RK3-TVD + WENO5. For this scheme the one-step is and the stability condition is therefore, 4 6 ( ( t, x O ( t ( x σ = + (4 t = K x ( ( (( GTE R3W5 ξ (, t = O ( x + ( x = O Now, for all schemes we proceed by to bild the spatial grid by letting vector {,d,d,,( d π} (5 (6 π dx =, k and we define the X = x x N x = (7 { X (, X (,, X ( N, X ( N } = (8 which has N points. As the bondary conditions are periodic, we will have to make the following identifications, ( = ( 3, ( = (, ( = (, ( = (, ( + = ( X X N X X N X X N X X N X N X To bild the grid in time, we first dt = rdx for some r satisfying the CFL (condition. Therefore we define N = ceil( τ dt where τ is the final time. So we going to iterate ntil the time t = N dt. Then, in general t τ bt at least. Here, all times steps are the same size dt. Finally, when compting the global trncation error, the only important thing is to always compte it at the same time t for all grid resoltions. Now, the scheme (LF, LW or RK3-WENO5 is applied ntil t is reached. Then the otpt is app. The grid resoltion dx, calclate what the tre soltion is at time t and compte the norm of the errors by sing i=,, K app ( ( E = max i i K tre ( ( app tre i= k (9 E = dx i i ( K = d app ( tre ( ( i= E x i i Here we iterate over the resoltion by varying k in order to get a convergence plot of the varios norms of the 469

4 errors vs. grid size. Lax-Friedrich. This scheme can be coded very efficiently sing matrices. We can see that the scheme can be rewritten as n+ n n n n j = j+ j ( r j+ ( r j + + = + + x x therefore taking into accont the periodic bondary conditions n+ r r + r 3 r 3 = K + r K K r + rk K r r + K Lax-Wendroff. Similarly this scheme can be implemented by sing r r r ( r ( + r r r ( r r ( r + ( r = r r ( r K r r ( r ( + r r n+ r 3 + r 3 K K K K RK3-TVD-WENO5. For each point on the grid, we need to compte two intermediate vales as follows. ( n n t WENO( n n K n ( ξ = FE ξ = ξ ξ (3 i i i i ( ( ( t WENO ( n 3 n FE FE 3 n ξ i = ξi + ξi = ξ i ξ i + ξi ( ( n ( n FE WENO( n ξ + i = ξ i + ξi = ξ i ξ i + ξi ( Therefore, the code for WENO5, when solving a conservative eqation of the form provided and f are n c >, define x. If ( i We can say that i ( ( + f = + f = (6 t x t x C. Then c( ( xt, : f ( ( xt, = 6, then = acts as the speed fnction. Then at the interior point (7 ( v, v,, v = (,,, 5 i i 3 i i i+ i+ 3 s = ( v v + v3 + ( v 4v + 3v3 (8 4 47

5 finally we obtain 3 s = ( v v + v4 + ( v v4 (9 4 3 s3 = ( v3 v4 + v5 + ( 3v3 4v4 + v5 ( ,, =,, ( α α α ( + ( + ( + 3 s s s3 (3 s α = α+ α + α3 (3 = α α α (33 s α ( w, w, w (,, 3 3 n WENO ( i = w( v 7v + v3 + w( v + 5v3+ v4 + w3( v3+ 5v4 v5 6 (34 3. Test Problems Example. We consider the condition ( x = cosx, then by sing the schemes we have: Lax-Friedrich. The convergence reslts are shows on Figre. (Convergence plot for ( xt, = cos( x t in Log-Log scale, advected by sing L-F scheme. log ( GTE at T is plotted against log ( h. The 5 grid resoltion was varied from dx = π to already visible at those resoltions. x = π 6. So the expected orders of convergence were Lax-Wendroff. The convergence reslts are shown on Figre. (Convergence plot for ( xt, = cos( x t in Log-Log scale, advected by sing L-W scheme. log ( GTE at T is plotted against log ( h. As 5 above, the grid resoltion was varied from dx = π to x = π. RK3-TVD + WENO5. The convergence reslts are shown on Figre 3. (Convergence plot for ( xt, = cos( x t in Log-Log scale, advected by sing L-W scheme. log ( GTE at T is plotted 4 against log ( h. As above, the grid resoltion was varied from dx = π to x = π. The norms errors behave as expected in the asymptotic behavior. This example nderlines the importance of letting the error go to whenever possible. Here, since the compter time became nreasonably long for machine Figre. Convergence plot for ( xt, cos( x t = with L-F scheme. 47

6 Figre. Convergence plot for ( xt, cos( x t = with L-W scheme. Figre 3. Convergence plot for ( xt, cos( x t = with RK3-TVD-WENO5 scheme. high grid resoltions, this was not possible. However, as sggested in class, one way to observe the order of the global trncation error (GTE is to increase the final time (for example T = 5 in order to let error in t accmlate and become more significant than the error in x. Example. Consider the box condition (, then we have Lax-Friedrich. The convergence reslts are shown on Figre 4. (Convergence plot for ( xt, = cos( x t in Log-Log scale, advected by sing L-W scheme. log ( GTE at T is plotted against log ( h. As 4 6 above, the grid resoltion was varied from dx = π to x = π. Therefore, since in deriving the LTE and GTE, we sed the Taylor series of the fnction ( xt,. Bt the box fnction being only C, we cannot expect ( xt, to eqal its Taylor series point wise, and thereforethe above reasoning does not apply. It is interesting to watch the box evolve when advected with a L-F scheme. This is presented in a series of snapshots pt together in Figre 5. (The tre soltion is plotted in red and the nmerical soltion is plotted in ble. Clearly, the schemes fails to captre the discontinities of the fnction, and its diffsive natre eventally leads to large errors near the discontinities. This explains why the L norm of the error cannot converge. This also explains why the L norm, which is simply the area between the tre soltion and the nmerical soltion, improves with the grid resoltion. xt, = box xt, Lax-Wendroff. The convergence reslts are shown on Figre 6. (Convergence plot for ( ( 47

7 Figre 4. Convergence plot for ( xt, box ( xt, = with L-F scheme. Figre 5. Snapshots for ( xt, box ( xt, scheme. = with L-F Figre 6. Convergence plot for ( xt, = box ( xt, with L-W scheme. in Log-Log scale, advected by sing L-W scheme. ( above, the grid resoltion was varied from clearly isn t convergence. log GTE at log h. As 5 6 dx = π to x = π. The slope for the L, since there T is plotted against ( 473

8 It is interesting to watch the box evolve when advected by a L-W scheme. This is presented in a series of snapshots pt together in Figre 7 (tre soltion in red and nmerical soltion in ble. This scheme also fails to captre the discontinities of the fnction, and its dispersive natre eventally leads to large errors near the discontinities, where wild oscillations appear. However, the overall shape of the box is better preserved than when advected with L-F, which provides a simple explanation as to why the rate of convergence of the L norm is slightly better. RK3-TVD-WENO5. The convergence reslts are shown on Figre 8. (Convergence plot for box in log-log scale, advected by sing RK3-TVD-WENO5 scheme. As above, the grid resoltion was varied from 5 dx = π to dx = π The final time was increased to T = 4. Therefore We almost get first order convergence in the L norm, which is mch better than with the previos two schemes. However, convergence is slower in the L norm and absent in the L norm. Again, it is interesting to watch the box evolve when advected by an RK3-TVD-WENO5 scheme. Althogh this scheme also fails to captre the discontinities of the fnction, the overall shape is mch better preserved than when advected with the previos two linear schemes. Althogh the L norm cannot converge becase of the large errors near the discontinities, those errors do not spread ot and pollte the soltion, which explains why the rate of convergence of the L norm is almost. 4. Semi-Lagrangian Approach to bild an interpolant ( The idea is to se varios vales of j L x and the get = L x c t. We se the formla for Lagrange interpolation in each case, here r =.5 and c =. For the linear interpola-, x,, therefore tion we se the two neighboring points ( x and ( from ( Figre 7. Snapshots for ( xt, box ( xt, scheme. = with L-W Figre 8. Convergence plot for ( xt, = box ( xt, with RK3-TVD-WENO5 scheme. 474

9 then and finally x x x x x x x x L x = l x + l x = + = + ( ( ( x x x x c t x x c c c L( x c t = + = + left, sing ( x,, ( x, and (, = c ( pwind method We can see that if we let r =.5 and c =, then =. Now, for the qadratic interpolation- x we have then finally ( ( ( ( ( x x ( x x ( x x ( x x ( x x ( x x L x = l x + l x + l x = + + L x c t = + c x t c t x ( ( 4 3 ( c c = Here, if we let r =.5 and ( 4 3 ( ( Beam-Warming method c =, then ( 4 3 ( = Now, with qa- 4 8 dratic interpolation-right and by sing ( x,, ( x, and ( 3, 3 therefore ( ( ( ( (35 (36 (37 (38 (39 (4 x we have ( x x ( x x ( x x ( x x ( x x ( x x L x = l x + l x + l x = + + finally we have c c L x c t = ( ( ( 3 3 c c = + + Lax-Wendroff method ( 3 ( 3 ( Now, if we let r =.5 and c =, then = ( 3 + ( We consider now a high-order interpolation, i.e. higher than qadratic. Proceeding in the same way in part above by sing 4 points, in order to get a cbic interpolant. Using ( x,, ( x,, ( x, and ( x3, 3 we have therefore, ( = ( + ( + ( + 3 3( ( x x ( x x ( x x ( x x ( x x ( x x L x l x l x l x l x = ( x x ( x x ( x x ( x x ( x x ( x x (4 (4 (43 (44 475

10 c c L x c t = c + 3 ( finally, c c = ( ( 3 + c c + ( ( Now, we can see the Stability as follows. Performing Von Nemann gives the following, ( ( 6 3 ( r r r k k k k k k + ( e 3e + 3 e 6 k k k k k k k k k k ( = + ( e + 6e 3 e + ( 3e 6 + 3e G k In order to find ot for which vales of r the scheme is stable, here we plotted ( G k for varios vales of r, ranging from r = and r = since that CFL condition for the other schemes was r, the reslts shown on Figre 9 (plot of G( k against k x for r = :. :, the top straight line corresponds to r = and therefore G( k r < G( k r for r < r pointwise, indicate that r ensres stability, since G( k for all k [, π x]. For the accracy we compte the LTE of the scheme, ξ (, t = ( ( xt, + ( xt, c x ( ( x xt, 6 ( x xt, 3 ( xt, ( x xt, c + ( 3 ( x t, 6 ( xt, + 3 ( x+t, 3 c + 3 ( ( x t, 3 ( x xt, + 3 ( xt, ( x+t,. 3 4 = ( t t tt ttt tttt c ( x ( x xxxx + ( xxxxx ( xxxxxx 6 x 5 c ( x + ( x + ( 6 xx xxxx xxxxxx 4 3 c ( x + ( ( x + ( 4 xxx xxxx xxxxx xxxxxx = t c x tt c xx ttt c xxx 6 = = = 3 3 c c tttt xxxx xxxxxx (( 3 ( 3. (45 (46 (47 (48 (49 (5 (5 = O x (5 476

11 3 ( r, we expect the GTE to be ( P. P. C. Alzate Therefore, if we se the CFL condition O t. We can see the reslts for ( x = cosx where the convergence is shown on Figre. Here the grid resoltion was varied from dx = π to dx = π. Finally, for ( x = box ( x t the convergence reslts are shown on Figre, where the grid resoltion was varied from dx = π to dx = π. Note. As for Lax-Friedrich and Lax-Wendroff, those reslts shold not be srprising, becase we cannot expect a method based on the Taylor series of a fnction to work for a C fnction. We can see the box evolve when advected by a 3rd order scheme. This is presented in a series of snapshots pt together in Figre. Althogh this scheme fails to captre the discontinities of the fnction, jst like the previos schemes, the oscillations created near the discontinities don t seem to grow very fast. Moreover, the overall shape of the box is (at least visally better preserved that when advected with L-F of L-W, which provides a simple explanation as to why the rate of convergence of the L norm is better than that with those schemes. Figre 9. Plot of G( k. Figre. Convergence plot for ( xt, cos( x t sing a 3rd order scheme. = advected 477

12 Figre. Convergence plot for ( xt, box ( x t = advected sing a 3rd order scheme. Acknowledgements Figre. Snapshots for ( xt, box ( x t 3rd order scheme. = advected by a I wold like to thank the referee for his valable sggestions that improved the presentation of this paper and my gratitde to Department of Mathematics of the Universidad Tecnológica de Pereira (Colombia and the grop GEDNOL. References [] Strikwerda, J.C. (989 Finite Difference Schemes and Partial Differential Eqations. Wadsworth & Brooks, USA. [] McRea, G.J. and Godin, W.R. (967 Nmerical Soltion of Atmospheric Diffsion for Chemically Reacting Flows. Jornal of Comptational Physics, 77, -4. [3] Morton, K.W. (98 Stability of Finite Difference Approximations to a Diffsion-Convection Eqation. International Jornal for Nmerical Methods in Engineering, 5, [4] Hndsdorfer, W. and Koren, B. (995 A Positive Finite-Difference Advection Scheme Applied on Locally Refined 478

13 Grids. Jornal of Comptational Physics, 7, [5] Canto, C. and Hssaini, M. (988 Spectral Methods in Flids Dynamics. Springer Series in Comptational Physics, Springer-Verlag, Berlin. [6] Mitchell, A.R. and Griffiths, D.F. (98 The Finite Difference Method in Partial Differential Eqations. John Wiley & Sons, Chichester. [7] Mickens, R.E. ( Applications of Nonstandard Finite Differences Schemes. World Scientific Pblishing, River Edge. [8] Dehghan, M. (5 On the Nmerical Soltion of the One-Dimensional Convection-Diffsion Eqation. Mathematical Problems in Engineering,,

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