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 ${W_t} = - {F_x} = - A{W_x}$, 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. These schemes generalize the Lax-Wendroff 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. (Courant-Friedrichs-Lewy) criterion of the Courant-number 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 CDC-3400 computer at the Tel Aviv University computation center.