Third and Fourth Order Accurate Schemes for Hyperbolic Equations of Conservation Law Form

It is shown that for quasi-linear hyperbolic systems of the conservation form W sub t = -(F sub x) = -A(W sub 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. This requirement is not fulfilled by any known physical systems of equations. These schemes generalize the Lax-Wendroff 2nd order one, and are written down explicitly. As found by Strang, odd order schemes are linearly unstable unless modified by adding a term containing the next higher space derivative. Thus stabilized, the schemes, both odd and even, can be made to meet the C.F.L. (Courant-Friedrichs-Lewy) criterion. 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.