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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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Third and fourth order accurate schemes for hyperbolic equations of conservation law form
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by Gideon Zwas and Saul Abarbanel PDF
Math. Comp. 25 (1971), 229-236 Request permission


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.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 229-236
  • MSC: Primary 65P05
  • DOI:
  • MathSciNet review: 0303766