On stability of numerical schemes via frozen coefficients and the magnetic induction equations

We study finite difference discretizations of initial boundary value problems for linear symmetric hyperbolic systems of equations in multiple space dimensions. The goal is to prove stability for SBP-SAT (Summation by Parts-Simultaneous Approximation Term) finite difference schemes for equations with variable coefficients. We show stability by providing a proof for the principle of frozen coefficients, i.e., showing that variable coefficient discretization is stable provided that all corresponding constant coefficient discretizations are stable. We apply this general result to the special case of magnetic induction equations and show that high order SBP-SAT schemes are energy stable even with boundary closures. Furthermore, we introduce a modified discretization of lower order terms and show that the discrete divergence of this scheme is bounded. The discrete divergence is shown to converge to zero under certain assumptions. Computations supporting our theoretical results are also presented.