Lie Symmetry Analysis of Shock Problems Using Generalized Functions

Lie symmetry analysis depends on the calculation of derivatives to find and use the symmetries. Solutions to hydrodynamic shock problems have discontinuities which mean that the usual derivative cannot be calculated. The solution is to use symmetry analysis in the smooth regions between shocks and stitch them together using the Hugoniot jump conditions, which must also satisfy the symmetry conditions. In this work we use generalized functions to express the solution and then calculate the derivatives using generalized derivatives. This provides a concise framework for analyzing shock problems. We demonstrate the method on the Noh and shock tube problems.