Third and fourth order accurate schemes for hyperbolic equations of conservation law form
Authors:
Gideon Zwas and Saul Abarbanel
Journal:
Math. Comp. 25 (1971), 229236
MSC:
Primary 65P05
MathSciNet review:
0303766
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Abstract: It is shown that for quasilinear hyperbolic systems of the conservation form , it is possible to build up relatively simple finitedifference numerical schemes accurate to 3rd and 4th order provided that the matrix A satisfies commutativity relations with its partialderivativematrices. These schemes generalize the LaxWendroff 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. (CourantFriedrichsLewy) criterion of the Courantnumber 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 CDC3400 computer at the Tel Aviv University computation center.
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 G. Strang, "Trigonometric polynomials and difference methods of maximum accuracy," J. Mathematical Phys., v. 41, 1962, p. 147.
 [9]
 A. K. Aziz, H. Hurwitz & H. M. Sternberg, "Energy transfer to a rigid piston under detonation loading," Phys. Fluids, v. 4, 1961, p. 380.
 [10]
 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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197103037664
PII:
S 00255718(1971)03037664
Keywords:
Quasilinear hyperbolic equations,
finitedifference schemes,
LaxWendroff methods,
numerical stability,
high order accuracy,
conservation laws
Article copyright:
© Copyright 1971 American Mathematical Society
