Shock calculations and the numerical solution of singular perturbation problems

Sod (1977) has compared different methods for calculating the solution of a shock tube problem. The problem involved the sudden removal of a membrane which separated a gas in two different states. Following the removal of the membrane, a moving shock, a rarefaction wave, and a contact discontinuity appeared. The methods compared include those of Hyman, Gudonov, Lax-Wendrof, McCormack, Rosanow, Glimm, and 'Shastra'. The results of the first five methods were very similar. The number of gridpoints was very large. In the present investigation, this problem was also considered, and experiments for simplified problems were conducted. Using the first five methods, a shock, a rarefaction wave, and travelling (contact) discontinuities were obtained. It was found that the method developed by Hyman (1979) performed best. Attention is also given to difference approximation for scalar equations, difference approximations for systems, and a nonlinear equation.