We consider Riemann data for the nonlinear wave system which result in a regular reflection with a subsonic state behind the reflected shock. The problem in self-similar coordinates leads to a system of mixed type and a free boundary value problem for the reflected shock and the solution in the subsonic region. We show existence of a solution in a neighborhood of the reflection point.

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We study an implicit spacetime Godunov method for a class of hyperbolic conservation laws known as Temple class systems. We establish the well-posedness of this method, a discrete entropy inequality, a property analogous to the total variation diminishing property of certain numerical schemes for scalar conservation laws, and, as a consequence, the convergence of the numerical method.

We prove local existence of a solution to a Riemann problem for the two-dimensional nonlinear wave system using the approach by Čanić, Keyfitz, Kim and Lieberman. We consider initial data resulting in weak regular shock reflection. By writing the problem in self-similar coordinates, we obtain a mixed type system and a free boundary problem for the subsonic flow and the position of the reflected shock. We reformulate this problem using a second order equation for density with mixed boundary conditions and an ordinary differential equation describing the location of the reflected shock. The main difficulty in the study of this second order problem for density is that the operator is degenerate elliptic along the sonic circle. We regularize and modify the operator, and we show existence of solutions to the regularized problems. We prove that the sequence of solutions to the regularized problems is precompact, and, using uniform local ellipticity, covering and diagonalization techniques, we obtain a local solution to the original problem.

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