Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Higher order accuracy finite difference algorithms for quasi-linear, conservation law hyperbolic systems

Authors: S. Abarbanel and D. Gottlieb
Journal: Math. Comp. 27 (1973), 505-523
MSC: Primary 65M10
MathSciNet review: 0334541
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An explicit algorithm that yields finite difference schemes of aly desired order of accuracy for solving quasi-linear hyperbolic systems of partial differential equations in several space dimensions is presented. These schemes are shown to be stable under certain conditions. The stability conditions in the one-dimensional case are derived for any order of accuracy. Analytic stability proofs for two and $ d\;(d > 2)$ space dimensions are also obtained up to and including third order accuracy. A conjecture is submitted for the highest accuracy schemes in the multi-dimensional cases. Numerical examples show that the above schemes have the stipulated accuracy and stability.

References [Enhancements On Off] (What's this?)

  • [1] J. von Neumann & R. D. Richtmyer, "A method for the numerical calculation of hydrodynamical shocks," J. Appl. Phys., v. 21, 1950, pp. 232-237. MR 12, 289. MR 0037613 (12:289b)
  • [2] P. D. Lax, "Weak solutions of nonlinear hyperbolic equations and their numerical computation," Comm. Pure Appl. Math., v. 7, 1954, pp. 159-193. MR 16, 524. MR 0066040 (16:524g)
  • [3] P. D. Lax & B. Wendroff, "Systems of conservation laws," Comm. Pure Appl. Math., v. 13, 1960, pp. 217-237. MR 22 #11523. MR 0120774 (22:11523)
  • [4] G. Strang, "Trigonometric polynomials and difference methods of maximum accuracy," J. Mathematical Phys., v. 41, 1962, p. 147.
  • [5] R. D. Richtmyer, A Survey of Difference Methods for Non-Steady Fluid Dynamics, NCAR Technical Notes 63-2, Boulder, Colorado, 1962.
  • [6] P. D. Lax & B. Wendroff, "Difference schemes for hyperbolic equations with high order of accuracy," Comm. Pure Appl. Math., v. 17, 1964, pp. 381-398. MR 30 #722. MR 0170484 (30:722)
  • [7] R. D. Richtmyer & K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed., Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1967. MR 36 #3515. MR 0220455 (36:3515)
  • [7] G. Strang, "Accurate partial difference methods. II: Non-linear problems," Numer. Math., v. 6, 1964, pp. 37-46. MR 29 #4215. MR 0166942 (29:4215)
  • [9] S. Z. Burstein & A. Mirin, "Third order difference methods for hyperbolic equations," J. Comput. Phys., v. 5, 1970, pp. 547-571. MR 43 #8255. MR 0282545 (43:8255)
  • [10] V. V. Rousanov, "On difference schemes of third order accuracy for non-linear hyperbolic systems," J. Comput. Phys., v. 5, 1970, pp. 507-516. MR 0275699 (43:1452)
  • [11] A. R. Gourlay & J. L. Morris, "Finite difference methods for non-linear hyperbolic systems," Math. Comp., v. 22, 1968, pp. 28-39. MR 36 #6163. MR 0223114 (36:6163)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65M10

Retrieve articles in all journals with MSC: 65M10

Additional Information

Keywords: Finite difference schemes, stability, accuracy, convergence, quasi-linear hyperbolic systems, conservation law forms, the Lax-Wendroff schemes, Richtmyer scheme, von Neumann condition
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society