Abstract
The implicit midpoint 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 predictorcorrector 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 secondorder integrators, such as the backward differentiation formula (BDF2) and the trapezoidal rule. The proposed algorithm is suitable for a wider class of nonlinear problems, which are linearised by Newton’s method and require the preservation of geometric properties in the numerical solution.
Original language  English 

Journal  Journal of Scientific Computing 
Early online date  21 May 2019 
DOIs  
Publication status  Published  21 May 2019 
Keywords
 Adaptive time integration
 Initial value problems
 Landau–Lifshitz–Gilbert equation
 Micromagnetics
 Predictorcorrector methods
 Runge–Kutta methods
Fingerprint
Dive into the research topics of 'An Adaptive Step Implicit Midpoint Rule for the Time Integration of Newton’s Linearisations of NonLinear Problems with Applications in Micromagnetics'. Together they form a unique fingerprint.Impacts

IFISS: A software package for teaching computational mathematics
David Silvester (Participant), Howard Elman (Participant) & Alison Ramage (Participant)
Impact: Awareness and understanding, Technological