Large time step shock-capturing techniques for scalar conservation laws

For a scalar conservation law $u_t\ = {f(u)}_x\ with f"$ of constant sign, the first order upwind difference scheme is a special case of Godonov''s method. The method is equivalent to solving a sequence of Riemann problems at each step and averaging the resulting solution over each cell in order to obtain the numerical solution at the next time level. The difference scheme is stable (and the solutions to the associated sequence of Riemann problems do not interact) provided the Courant number $\nu$ is less than 1. By allowing and explicitly handling such interactions, it is possible to obtain a generalized method which is stable for $\nu$ much larger than 1. In many cases the resulting solution is considerably more accurate than solutions obtained by other numerical methods. In particular, shocks can be correctly computed with virtually no smearing. The generalized method is rather unorthodox and still has some problems associated with it. Nonetheless, preliminary results are quite encouraging.