Hyperbolic Conservation Laws with Source Terms: Errors of the Shock Location

We investigate the error of the shock location which occurs in numerical solutions of hyperbolic conservation laws with source terms. For our theoretical analysis we consider a scalar Riemann problem. We compute its solution using a splitting method. This means that in each time step the homogeneous conservation law and an ODE (modeling the source term) are solved separately. We show that the local error of the shock location can be considered to consist of two parts: one part introduced by the splitting and another occurring because of smeared-out shock profiles. Numerical examples show that these error-estimates can be used to adapt the step size so that the error of the shock location remains sufficiently small. The numerical examples include one-dimensional systems