Typical Technological Trickery

Publication: Adaptive-step implicit-midpoint-rule for time integration

24 May 2019

The main paper from my PhD has finally made it through the publication process, and is available here (it’s open access so you don’t need an expensive subscription to read it).

The abstract is:

The implicit mid-point rule is a Runge–Kutta numerical integrator for the solution of initial value problems, which possesses important properties that are relevant in micromagnetic simulations based on the Landau–Lifshitz–Gilbert equation, because it conserves the magnetization length and accurately reproduces the energy balance (i.e. preserves the geometric properties of the solution). We present an adaptive step size version of the integrator by introducing a suitable local truncation error estimator in the context of a predictor-corrector scheme. We demonstrate on a number of relevant examples that the selected step sizes in the adaptive algorithm are comparable to the widely used adaptive second-order integrators, such as the backward differentiation formula (BDF2) and the trapezoidal rule. The proposed algorithm is suitable for a wider class of non-linear problems, which are linearised by Newton’s method and require the preservation of geometric properties in the numerical solution.

In plain English: I took an algorithm which has some useful properties for simulating magnetic materials and found a way to make it faster without losing those properties. Hopefully it will turn out to be useful in other areas of computational physics too, but I didn’t have time to look into that.