Noh's constant-velocity shock problem revisited

We present the solutions to Noh's shock tube problem in planar, cylindrical, and spherical geometry. This problem has the well-deserved reputation of being challenging to numerical methods. Since the gas is initially cold there are infinitely many reflections of the shock between the fixed wall and the piston as the piston moves with constant velocity towards the wall. An implicit adaptive grid algorithm allows us, for the first time, to generate the complete solutions in these three geometries. We discuss them in detail, in particular follow the shock over many reflections, and perform numerical consistency checks. For the planar case the exact analytical solution is derived, and the numerical error in all physical quantities is found to be less than 1% on a 100 grid-point computational domain. For the converging geometries an approximate analytical theory is presented, and the deviations between the theory and the numerical results are found to be less than 10% on the same domain. A substantial part of this total error is due to errors in the approximate analytical results. We discuss the physics of the shock reflection in the three geometries, and analyze in particular the finite amount of entropy that is produced after the the first shock reflection. In an appendix we present some details of our code and demonstrate that the adaptive grid permits us to carry out computations of extreme precision. The reliability of our solutions in all three geometries allows them to become demanding tests for 2D and 3D codes.