Heterogeneous Multiscale Methods for the Landau–Lifshitz Equation

Abstract

In this paper, we present a finite difference Heterogeneous Multiscale Method for the Landau–Lifshitz equation with a highly oscillatory diffusion coefficient. The approach combines a higher order discretization and artificial damping in the so-called micro problem to obtain an efficient implementation. The influence of different parameters on the resulting approximation error is discussed. Further important factors that are taken into account are the choice of time integrator and the initial data for the micro problem which has to be set appropriately to get a consistent scheme. Numerical examples in one and two space dimensions and for both periodic as well as more general coefficients are given to demonstrate the functionality of the approach.