×

zbMATH — the first resource for mathematics

A fourth-order central Runge-Kutta scheme for hyperbolic conservation laws. (English) Zbl 1200.65074
Summary: A new formulation for a central scheme recently introduced by A. A. I. Peer et al. [Appl. Numer. Math. 58, No. 5, 674–688 (2008; Zbl 1138.65075)] is presented. It is based on staggered grids. For this work, first a time discretization is carried out, followed by a space discretization. Spatial accuracy is obtained using piecewise cubic polynomial and fourth-order numerical derivatives. Time accuracy is obtained applying a Runge-Kutta scheme. The scheme proposed in this work has a simpler structure than the central scheme developed in [loc. cit.]. Several standard one-dimensional test cases are used to verify high-order accuracy, nonoscillatory behavior, and good resolution properties for smooth and discontinuous solutions.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Simpson, Nonmonotone chemotactic invasion: high-resolution simulation, phase plane analysis and new benchmark problems, J Comput Phys 225 pp 6– (2007) · Zbl 1201.92024
[2] Russo, Trends and applications of mathematics to mechanics: STAMM 2002 pp 225– (2005)
[3] Gabetta, Central schemes for hydrodynamical limits of discrete-velocity kinetic models, Trans Theo Stat Phys 29 pp 465– (2000) · Zbl 1017.82044
[4] Balbas, Non-oscillatory central schemes for one-and two-dimensional MHD equations. II: High-order semi-discrete schemes, SIAM J Sci Comput 28 pp 533– (2006)
[5] Dehghan, Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Mathematics Comput Simul 71 pp 16– (2006) · Zbl 1089.65085
[6] Dehghan, On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer Methods Partial Differential Eq 21 pp 24– (2005)
[7] LeVeque, Cambridge Texts in Applied Mathematics (2002)
[8] Liu, Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer Math 79 pp 379– (1998) · Zbl 0906.65093
[9] Nessyahu, Non-oscillatory central differencing for hyperbolic conservation laws, J Comput Phys 87 pp 408– (1990) · Zbl 0697.65068
[10] Peer, A new fourth-order non-oscillatory central scheme for hyperbolic conservation laws, Appl Numer Math 58 pp 674– (2008) · Zbl 1138.65075
[11] Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Math Sb 47 pp 271– (1959) · Zbl 0171.46204
[12] Friedrichs, Systems of conservation equations with a convex extension, Proc Nat Acad Sci 68 pp 1686– (1971) · Zbl 0229.35061
[13] Jiang, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J Sci Comput 19 pp 1892– (1998) · Zbl 0914.65095
[14] Liu, Non-oscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I, SIAM J Numer Anal 33 pp 760– (1996) · Zbl 0859.65091
[15] Romano, Numerical solution for hydrodynamical models of semiconductors, Math Models Methods Appl Sci 10 pp 1099– (2000) · Zbl 1012.82027 · doi:10.1142/S0218202500000550
[16] Pareschi, Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms, SIAM J Numer Anal 39 pp 1395– (2001) · Zbl 1020.65048
[17] Liotta, Central schemes for balance laws of relaxation type, SIAM J Numer Anal 38 pp 1337– (2000) · Zbl 0982.65093
[18] Bryson, High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton-Jacobi equations, J Comput Phys 189 pp 63– (2003) · Zbl 1027.65126
[19] Pareschi, Central Runge Kutta schemes for conservation laws, SIAM J Sci Comput 26 pp 979– (2005) · Zbl 1077.65094
[20] Zennaro, Natural continuous extensions of Runge-Kutta methods, Math Comp 46 pp 119– (1986) · Zbl 0608.65043
[21] Kurganov, Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J Sci Comput 23 pp 707– (2001) · Zbl 0998.65091
[22] Bianco, High-order central schemes for hyperbolic systems of conservation laws, SIAM J Sci Comput 21 pp 294– (1999) · Zbl 0930.65096
[23] Harten, Uniformly high-order accurate nonoscillatory schemes, I, SIAM J Numer Anal 24 pp 279– (1987) · Zbl 0627.65102
[24] A. A. I. Peer, A. Gopaul, M. Z. Dauhoo, and M. Bhuruth, New high-order ENO reconstruction for hyperbolic conservation laws, Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE-2005 Alicante, June 27-30, 2005, pp. 446-455. · Zbl 1209.65087
[25] J. C. Strikwerda, Finite difference schemes and partial differential equations, SIAM, 2004. · Zbl 1071.65118
[26] Kurganov, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer Math 88 pp 683– (2001) · Zbl 0987.65090
[27] Kurganov, New high-resolution central schemes for non-linear conservation laws and convection-diffusion equations, J Comput Phys 160 pp 241– (2000) · Zbl 0987.65085
[28] Levy, Central WENO schemes for hyperbolic systems of conservation laws, Math Model Numer Anal 33 pp 547– (1999) · Zbl 0938.65110
[29] Levy, A fourth-order central WENO schemes for multidimensional hyperbolic systems of conservation laws, SIAM J Sci Comput 24 pp 456– (2002) · Zbl 1014.65079
[30] Levy, Compact central WENO schemes for multidimensional conservation laws, SIAM J Sci Comput 22 pp 656– (2000) · Zbl 0967.65089
[31] Sod, A survey of several finite difference methods for systems of non-linear hyperbolic conservation laws, J Comput Phys 27 pp 1– (1978) · Zbl 0387.76063
[32] Lax, Weak solutions of non-linear hyperbolic equations and their numerical computation, Comm Pure Appl Math 7 pp 159– (1954) · Zbl 0055.19404
[33] Shu, Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J Comput Phys 83 pp 32– (1989) · Zbl 0674.65061
[34] Woodward, The numerical solution of two-dimensional fluid flow with strong shocks, J Comput Phys 54 pp 115– (1988)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.