Stable finite difference schemes for the magnetic induction equation with Hall effect

Abstract

We consider a sub-model of the Hall-MHD equations: the so-called magnetic induction equations with Hall effect. These equations are non-linear and include third-order spatial and spatio-temporal mixed derivatives. We show that the energy of the solutions is bounded and design finite difference schemes that preserve the energy bounds for the continuous problem. We design both divergence preserving schemes and schemes with bounded divergence. We present a set of numerical experiments that demonstrate the robustness of the proposed schemes.