The Guderley problem revisited

The self-similar converging-diverging shock wave problem introduced by Guderley in 1942 has been the source of considerable mathematical and physical interest. We investigate a novel application of the Guderley solution as a unique and challenging code verification test problem for compressible flow algorithms; this effort requires a unified understanding of the problem's mathematical and computational subtleties. Hence, we review the simplifications and group invariance properties that reduce the compressible flow equations for a polytropic gas to two coupled nonlinear eigenvalue problems: the first for the similarity exponent in the converging regime, and the second for a trajectory multiplier in the diverging regime. The information we provide, together with previously published material, gives a complete description of the computational steps required to construct a semi-analytic Guderley solution. We employ the problem in a quantitative code verification analysis of a cell-centred, finite volume, Eulerian compressible flow algorithm. Lastly, in appended material, we introduce a new approximation for the similarity exponent, which may prove useful in the future construction of certain semi-analytic Guderley solutions.