×

An analysis on induced numerical oscillations by Lax-Friedrichs scheme. (English) Zbl 1375.65112

Summary: Induced numerical oscillations in the computed solution by monotone schemes for hyperbolic conservation laws has been a focus of recent studies. In this work using a local maximum principle, the monotone stable Lax-Friedrichs scheme is investigated to explore the cause of induced local oscillations in the computed solution. It expounds upon that LxF scheme is locally unstable and therefore exhibits induced such oscillations. The carried out analysis gives a deeper insight to characterize the type of data extrema and the solution region which cause local oscillations. Numerical results for benchmark problems are also given to support the theoretical claims.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Software:

HE-E1GODF; SHASTA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boris, J.P., Book, D.L.: Flux-corrected transport. i. shasta, a fluid transport algorithm that works. J. Comput. Phys. 11, 38-69 (1973) · Zbl 0251.76004 · doi:10.1016/0021-9991(73)90147-2
[2] Breuß, M.: The correct use of the lax-friedrichs method. ESAIM: Mathematical Modelling and Numerical Analysis, M2AN, pp. 519-540 (2004) · Zbl 1077.65089
[3] Breuß, M.: An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws. ESAIM Math. Model. Num. Anal. 39(5), 965-994 (2005) · Zbl 1077.35089 · doi:10.1051/m2an:2005042
[4] Crandall, M.G., Majda, A.: Monotone difference approximations for scalar conservation laws. Math. Comput. 34, 1-21 (1980) · Zbl 0423.65052 · doi:10.1090/S0025-5718-1980-0551288-3
[5] Dubey, R.K.: Flux limited schemes: their classification and accuracy based on total variation stability regions. Appl. Math. Comput. 224, 325-336 (2013) · Zbl 1334.65130
[6] Dubey, R.K.: Total variation stability and second-order accuracy at extrema. Electron. J. Diff. Eqns. 20, 53-63 (2013) · Zbl 1292.65099
[7] Dubey, R.K., Biswas, B., Gupta, V.: Local maximum principle satisfying high order non-oscillatory schemes. Int. J. Num. Methods Fluids. doi:10.1002/fld.4202 · Zbl 0055.19404
[8] Laney, C.B., Caughey, D.: Extremum control. II—Semidiscrete approximations to conservation laws. Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics (1991)
[9] Laney, Culbert B.: Computational Gasdynamics. Cambridge University Press, Cambridge (1998) · Zbl 0947.76001 · doi:10.1017/CBO9780511605604
[10] Lax, Peter D.: Weak solutions of non-linear hyperbolic equations and their numerical approximation. Commun. Pure Appl. Math. 7, 159-193 (1954) · Zbl 0055.19404 · doi:10.1002/cpa.3160070112
[11] Lax, P.D., Wendroff, B.: Systems of conservation laws. Comm. Pure Appl. Math. 13, 217-237 (1960) · Zbl 0152.44802 · doi:10.1002/cpa.3160130205
[12] Lefloach, P.G., Liu, J.G.: Generalized monotone schemes, discrete paths of extrema and discrete entropy conditions. Math. Comput. 68, 1025-1055 (1999) · Zbl 0915.35069 · doi:10.1090/S0025-5718-99-01062-5
[13] LeVeque, R.J.: Numerical Methods for Conservation Laws. Lectures in mathematics ETH Zürich. Birkhäuser Basel, 2nd edn (1992) · Zbl 0847.65053
[14] Li, J., Tang, H., Warnecke, G., Zhang, L.: Local oscillations in finite difference solutions of hyperbolic conservation laws. Math. Comput. 78, 1997-2018 (2009) · Zbl 1198.65169 · doi:10.1090/S0025-5718-09-02219-4
[15] Li, J., Yang, Z.: Heuristic modified equation analysis of oscillations in numerical solutions of conservation laws. SIAM J. Num. Anal. 49, 2386-2406 (2011) · Zbl 1248.65094 · doi:10.1137/110822591
[16] Nessyahu, H., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408-436 (1990) · Zbl 0697.65068 · doi:10.1016/0021-9991(90)90260-8
[17] Osher, S., Tadmor, E.: On the convergence of difference approximations to scalar conservation laws. Math. Comput. 50, 19-51 (1988) · Zbl 0637.65091 · doi:10.1090/S0025-5718-1988-0917817-X
[18] Sanders, R.: On the convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput. 40, 91-106 (1983) · Zbl 0533.65061 · doi:10.1090/S0025-5718-1983-0679435-6
[19] Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. Siam J. Num. Anal. 21(5), 995-1011 (1984) · Zbl 0565.65048 · doi:10.1137/0721062
[20] Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn. Springer, Berlin (2009) · Zbl 1227.76006 · doi:10.1007/b79761
[21] Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091-3120 (2010) · Zbl 1187.65096 · doi:10.1016/j.jcp.2009.12.030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.