Seven-point difference schemes for hyperbolic equations

Author:
Avishai Livne

Journal:
Math. Comp. **29** (1975), 425-433

MSC:
Primary 65M05

DOI:
https://doi.org/10.1090/S0025-5718-1975-0398114-1

MathSciNet review:
0398114

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Abstract: A necessary and sufficient condition is given for all hyperbolic difference schemes that use up to nine mesh points to be of second-order accuracy. We also construct a new difference scheme for two-dimensional hyperbolic systems of conservation laws. The scheme is of second-order accuracy and requires knowledge of only seven mesh points. A stability condition is obtained and is utilized in numerical computations.

**[1]**B. EILON, D. GOTTLIEB & G. ZWAS, "Numerical stabilizers and computing time for second-order accurate schemes,"*J. Computational Phys.*, v. 9, 1972, pp. 387-397. MR**45**#9517. MR**0300471 (45:9517)****[2]**A. R. GOURLAY & J. Ll. MORRIS, "A multistep formulation of the optimized Lax-Wendroff method for nonlinear hyperbolic systems in two space variables,"*Math. Comp.*, v. 22 1968, pp. 715-719. MR**40**#5156. MR**0251931 (40:5156)****[3]**H. O, KREISS, "Über die Stabilitätsdefinition für Differenzengleichungen, die partielle Differentialgleichungen Approximieren,"*Nordisk Tidskr. Informationsbehandling*(*BIT*), v. 2, 1962, pp. 153-181. MR**0165712 (29:2992)****[4]**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)****[5]**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)****[6]**R. W. MacCORMACK,*Numerical Solution of the Interaction of a Shock Wave with a Laminar Boundary Layer*, Proc. 2nd Internat. Conf. on Numerical Methods in Fluid Dynamics (M. Holt, Editor), Springer-Verlag Lecture Notes in Phys., 1970.**[7]**R. D. RICHTMYER,*A Survey of Difference Methods for Nonsteady Fluid Dynamics*, N.C.A.R. Technical Notes, 1963, pp. 2-63.**[8]**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)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0398114-1

Keywords:
Quasi-linear hyperbolic equations,
finite-difference schemes,
Lax-Wendroff,
seven points,
consistency,
stability conservation laws

Article copyright:
© Copyright 1975
American Mathematical Society