Heterogeneous multiscale methods for the Landau-Lifshitz equation. (English) Zbl 1503.65175
Summary: 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.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 82D40 | Statistical mechanics of magnetic materials |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 78A25 | Electromagnetic theory (general) |
| 35Q60 | PDEs in connection with optics and electromagnetic theory |
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