Error Estimators for the Position of Discontinuities in Hyperbolic Conservation Laws with Source Terms which are solved using Operator Splitting

When computing numerical solutions of hyperbolic conservation laws with source terms, one may obtain spurious solutions --- these are unphysical solutions that only occur in numerics such as shock waves moving with wrong speeds, cf. [7], [2], [1], [10], [3]. Therefore it is important to know how errors of the location of a discontinuity can be controlled. To derive appropriate error-estimates and to use them to control such errors, is the aim of our investigations in this paper. We restrict our considerations to numerical solutions which are computed by using a splitting method. In splitting methods, the homogeneous conservation law and an ordinary differential equation (modelling the source term) are solved separately in each time step. Firstly, we derive error-estimates for the scalar Riemann problem. The analysis shows that the local error of the location of a discontinuity mainly consists of two parts. The first part is introduced by the splitting and the second part is due to smearing of the discontinuity. Next, these error-estimates are used to construct an adaptation of the step size so that the error of the location of the discontinuity remains sufficiently small. The adaptation is applied to several examples, which are a scalar problem, a simplified combustion model, and the one-dimensional inviscid reacting compressible Euler equations. All the examples show that the adaptation based on the derived error-estimates works well. The theory can also be extended to planar two-dimensional problems