On initial-value and self-similar solutions of the compressible Euler equations

We examine numerically the issue of convergence for initial‐value solutions and similarity solutions of the compressible Euler equations in two dimensions in the presence of vortex sheets (slip lines). We consider the problem of a normal shock wave impacting an inclined density discontinuity in the presence of a solid boundary. Two solution techniques are examined: the first solves the Euler equations by a Godunov method as an initial‐value problem and the second as a boundary value problem, after invoking self‐similarity. Our results indicate nonconvergence of the initial‐value calculation at fixed time, with increasing spatial‐temporal resolution. The similarity solution appears to converge to the weak ‘zero‐temperature’ solution of the Euler equations in the presence of the slip line. Some speculations on the geometric character of solutions of the initial‐value problem are presented.