Convergence of an implicit-explicit midpoint scheme for computational micromagnetics

TL;DR

This paper introduces a second-order convergent numerical scheme combining implicit and explicit methods for solving the Landau-Lifschitz-Gilbert equation in micromagnetics, with proven convergence and supporting numerical results.

Contribution

It develops a novel implicit-explicit midpoint scheme for LLG that achieves second-order accuracy and unconditional convergence, improving computational efficiency and reliability.

Findings

Proves unconditional convergence of the scheme.

Demonstrates second-order accuracy through numerical experiments.

Validates theoretical results with computational tests.

Abstract

Based on lowest-order finite elements in space, we consider the numerical integration of the Landau-Lifschitz-Gilbert equation (LLG). The dynamics of LLG is driven by the so-called effective field which usually consists of the exchange field, the external field, and lower-order contributions such as the stray field. The latter requires the solution of an additional partial differential equation in full space. Following Bartels and Prohl (2006) (Convergence of an implicit finite element method for the Landau-Lifschitz-Gilbert equation. SIAM J. Numer. Anal. 44), we employ the implicit midpoint rule to treat the exchange field. However, in order to treat the lower-order terms effectively, we combine the midpoint rule with an explicit Adams-Bashforth scheme. The resulting integrator is formally of second-order in time, and we prove unconditional convergence towards a weak solution of LLG. Numerical experiments underpin the theoretical findings.