An iterative Riemann solver for systems of hyperbolic conservation laws, with application to hyperelastic solid mechanics

In this paper, we present a general iterative method for the solution of the Riemann problem for hyperbolic systems of PDEs. The method is based on the multiple shooting method for free boundary value problems. We demonstrate the method by solving one-dimensional Riemann problems for hyperelastic solid mechanics. Even for conditions representative of routine laboratory conditions and military ballistics, dramatic differences are seen between the exact and approximate Riemann solution. The greatest discrepancy arises from misallocation of energy between compressional and thermal modes by the approximate solver, resulting in nonphysical entropy and temperature estimates. Several pathological conditions arise in common practice and modifications to the method to handle these are discussed. These include points where genuine nonlinearity is lost, degeneracies, and eigenvector deficiencies that occur upon melting.