# The extrapolated explicit midpoint scheme for variable order and step size controlled integration of the Landau-Lifschitz-Gilbert equation

## TL;DR

This paper introduces an efficient, adaptive higher-order integration scheme for the Landau-Lifschitz-Gilbert equation, utilizing extrapolation of the explicit midpoint rule and a piecewise stray field approximation to improve computational efficiency.

## Contribution

It presents a novel extrapolated explicit midpoint scheme with adaptive order and step size control for LLG equation integration, incorporating a new stray field approximation method.

## Key findings

- The scheme achieves higher efficiency compared to traditional methods.
- Numerical experiments confirm the method's accuracy and computational savings.
- Performance improves when stray field computation dominates the cost.

## Abstract

A practical and efficient scheme for the higher order integration of the Landau-Lifschitz-Gilbert (LLG) equation is presented. The method is based on extrapolation of the two-step explicit midpoint rule and incorporates adaptive time step and order selection. We make use of a piecewise time-linear stray field approximation to reduce the necessary work per time step. The approximation to the interpolated operator is embedded into the extrapolation process to keep in step with the hierarchic order structure of the scheme. We verify the approach by means of numerical experiments on a standardized NIST problem and compare with a higher order embedded Runge-Kutta formula. The efficiency of the presented approach increases when the stray field computation takes a larger portion of the costs for the effective field evaluation.