Riemann problems for nonstrictly hyperbolic 2×2 systems of conservation laws

Abstract

The Riemann problem is solved for 2 × 2 2 \times 2 systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem is complicated, with new types of shock waves, and new singularities in the dependence of the solution on the initial data. Several ideas are introduced to help organize and clarify the new phenomena.