No abstract

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. Keywords Initial value problems • Runge-Kutta methods • Adaptive time integration • Predictor-corrector methods • Micromagnetics • Landau-Lifshitz-Gilbert equation 1 Background and Context Initial value problems (IVP) arise in mathematical models of many important physical and engineering processes and phenomena. They either appear as stand-alone problems (as ordinary differential equations (ODEs), where the unknown functions depend only on a single variable, for example describing the motion in classical mechanics or chemical reactions), or in the context of applying the method of lines to partial differential equations (PDE), i.e. the problems where unknown functions have both spatial and time variation [1, p. 8]. The solutions of such problems frequently exhibit multiple spatio-temporal scales. In such cases implicit solvers allow the deployment of larger step sizes, while maintaining stability. Efficient integrators should also involve adaptivity [2, p. 73,74], which can be realised both

No abstract

No abstract