Third and fourth order accurate schemes for hyperbolic equations of conservation law form
Authors:
Gideon Zwas and Saul Abarbanel
Journal:
Math. Comp. 25 (1971), 229-236
MSC:
Primary 65P05
DOI:
https://doi.org/10.1090/S0025-5718-1971-0303766-4
MathSciNet review:
0303766
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that for quasi-linear hyperbolic systems of the conservation form ${W_t} = - {F_x} = - A{W_x}$, it is possible to build up relatively simple finite-difference numerical schemes accurate to 3rd and 4th order provided that the matrix A satisfies commutativity relations with its partial-derivative-matrices. These schemes generalize the Lax-Wendroff 2nd order scheme, and are written down explicitly. As found by Strang [8] odd order schemes are linearly unstable, unless modified by adding a term containing the next higher space derivative or, alternatively, by rewriting the zeroth term as an average of the correct order. Thus stabilized, the schemes, both odd and even, can be made to meet the C.F.L. (Courant-Friedrichs-Lewy) criterion of the Courant-number being less or equal to unity. Numerical calculations were made with a 3rd order and a 4th order scheme for scalar equations with continuous and discontinuous solutions. The results are compared with analytic solutions and the predicted improvement is verified. The computation reported on here was carried out on the CDC-3400 computer at the Tel Aviv University computation center.
- Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI https://doi.org/10.1002/cpa.3160130205
- Peter D. Lax and Burton Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17 (1964), 381–398. MR 170484, DOI https://doi.org/10.1002/cpa.3160170311 R. D. Richtmyer, A Survey of Difference Methods for Non-Steady Fluid Dynamics, NCAR Technical Notes 63-2, Boulder, Colorado, 1962.
- S. Abarbanel and G. Zwas, An iterative finite-difference method for hyperbolic systems, Math. Comp. 23 (1969), 549–565. MR 247783, DOI https://doi.org/10.1090/S0025-5718-1969-0247783-2
- A. R. Gourlay and J. Ll. Morris, Finite difference methods for nonlinear hyperbolic systems, Math. Comp. 22 (1968), 28–39. MR 223114, DOI https://doi.org/10.1090/S0025-5718-1968-0223114-8
- Peter D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Mathematical Phys. 5 (1964), 611–613. MR 165243, DOI https://doi.org/10.1063/1.1704154 S. Z. Burstein & A. Mirin, Third Order Accurate Difference Methods for Hyperbolic Equations, Courant Institute Report, NYO-1480-136, Dec. 1969. G. Strang, "Trigonometric polynomials and difference methods of maximum accuracy," J. Mathematical Phys., v. 41, 1962, p. 147. A. K. Aziz, H. Hurwitz & H. M. Sternberg, "Energy transfer to a rigid piston under detonation loading," Phys. Fluids, v. 4, 1961, p. 380. S. Abarbanel & G. Zwas, "The motion of shock waves and products of detonation confined between a wall and a rigid piston," J. Math. Anal. Appl., v. 28, 1969, p. 517.
Retrieve articles in Mathematics of Computation with MSC: 65P05
Retrieve articles in all journals with MSC: 65P05
Additional Information
Keywords:
Quasi-linear hyperbolic equations,
finite-difference schemes,
Lax-Wendroff methods,
numerical stability,
high order accuracy,
conservation laws
Article copyright:
© Copyright 1971
American Mathematical Society