Solving Riemann problems with a topological tool

Abstract

In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and rarefaction waves. In the present paper, we construct, in detail, the three-dimensional wave manifold for a system of two conservation laws with quadratic flux functions. Using adapted coordinates, we derive explicit formulae for important surfaces and curves within the wave manifold and display them graphically. The surfaces subdivide the manifold into regions according to shock type, such as ones corresponding to the Lax admissibility criterion. The curves parametrize rarefaction, shock, and composite waves appearing in contiguous wave patterns. Whereas wave curves overlap in state space, they are disentangled within the wave manifold. We solve a Riemann problem by constructing a wave curve associated with the slow characteristic speed family, generating a surface from it using shock curves, and intersecting this surface with a fast family wave curve. This construction is applied to solve Riemann problems for several illustrative cases.