Finite-volume schemes for Friedrichs systems with involutions

Abstract

In applications solutions of systems of hyperbolic balance laws often have to satisfy additional side conditions. We consider initial value problems for the general class of Friedrichs systems where the solutions are constrained by differential conditions given in the form of involutions. These occur in particular in electrodynamics, electro- and magnetohydrodynamics as well as in elastodynamics. Neglecting the involution on the discrete level typically leads to instabilities.To overcome this problem in electrodynamical applications it has been suggested in Munz et al. (2000) to solve an extended system. Here we suggest an extended formulation to the general class of constrained Friedrichs systems. It is proven for explicit Finite-Volume schemes that the discrete solution of the extended system converges to the weak solution of the original system for vanishing discretization and extension parameter under appropriate scalings. Moreover we show that the involution is weakly satisfied in the limit. The proofs rely on a reformulation of the extension as a relaxation-type approximation and careful use of the convergence theory for finite-volume methods for systems of Friedrichs type. Numerical experiments illustrate our analytical results.