Abstract : In this paper a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow band of 2-3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the Courant-Friedrichs-Levy criterion. Test calculations of one dimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces. The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Neumann and elaborated by other workers in this field.
[1]
R. Courant,et al.
Über die partiellen Differenzengleichungen der mathematischen Physik
,
1928
.
[2]
R. D. Richtmyer,et al.
A Method for the Numerical Calculation of Hydrodynamic Shocks
,
1950
.
[3]
Fritz John,et al.
On integration of parabolic equations by difference methods: I. Linear and quasi‐linear equations for the infinite interval
,
1952
.
[4]
P. Lax.
Weak solutions of nonlinear hyperbolic equations and their numerical computation
,
1954
.
[5]
R. D. Richtmyer,et al.
Survey of the stability of linear finite difference equations
,
1956
.
[6]
Francis H. Harlow,et al.
Hydrodynamic Problems Involving Large Fluid Distortions
,
1957,
JACM.
[7]
P. Lax.
Hyperbolic systems of conservation laws II
,
1957
.
[8]
P. Lax.
The scope of the energy method
,
1960
.