The numerical solution of the Cauchy problem for two-dimensional Riemann problem with three-piece initial function

This is a note about the solution of Riemann-type initial value problems for a single conservation law in two space dimensions. By using specific examples we exhibit new phenomena in the qualitative structure of solutions of hyperbolic conservation laws. There are good existence and uniqueness theories for the solution of a single conservation law in several spacedimensions based upon the viscosity method , but the method gives one little insight into the qualitative structure of the discontinuity set of solutions. In one space dimention , the qualitative shock phenomena which occur have been described by Gelfand and Ballou and fit into the framework of regularity theorems . The examples which we calculate here display new kinds of "rarefaction" phenomena which occur in the solution of equations in two space dimentions.