Higher Order Accuracy Finite Difference Algorithms for Quasi - Linear , Conservation Law Hyperbolic Systems

An explicit algorithm that yields finite difference schemes of any 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. Introduction. The task of solving numerically the equations of gas dynamics has given rise in the last 25 years to the search for finite difference algorithms of increasing accuracy and efficiency. The pioneering work in the late 1940's of von Neumann and Richtmyer [1] on the one-dimensional case led to work of Lax [2], Lax and Wendroff [3], Strang [4] and Richtmyer [5]. By the mid-sixties, the problem of constructing stable 2nd order algorithms in two space dimensions was solved by Lax and Wendroff [6], Richtmyer [7, p. 361] and Strang [4], [8]. Burstein and Mirin [9] and Rusanov [10] then solved the 3rd order accuracy case while Strang's [4] work included arbitrary order of accuracy for a linear system in one space dimension. In the present paper, the following results are presented: (1) An explicit algorithm that yields finite difference equations that approximate the quasi-linear hyperbolic system to any desired accuracy and for arbitrary number of space dimensions. (2) Analytic stability proofs and criteria of the above-mentioned algorithms in the case of one dimension, for arbitrary order of accuracy. (3) Analytic stability proofs in the 2 and d id > 2) dimensional cases up to and including 3rd order accuracy with sufficient stability conditions. (4) Numerical examples are carried out for a one-dimensional 2X2 system and a two-dimensional 2X2 system. The computed values are compared with analytic solutions and are shown to have the stipulated accuracies (4th order for the 1-D case and 3rd order for the 2-D case). " Received March 22, 1972. AMS (MOS) subject classifications (1970). Primary 65M05, 65M10.