Construction of adaptive exponential multi-operator splitting methods
Othmar Koch, Koray Acar, Winfried Auzinger, Daniel Hoffmann, Friedrich Kupka, Benedikt Moser
TL;DR
The paper addresses adaptive time-stepping for magnetohydrodynamics by designing second-order, multi-operator splittings into 3–4 operators with positive coefficients to respect physical constraints. It introduces Milne-pair–based local error estimators and optimizes operator coefficients to minimize the leading local error term $C\tau^{p+1}$, validating the approach on a Burgers equation surrogate that exhibits shock-like dynamics. Key contributions include explicit coefficient sets for four- and three-operator splittings, both under positivity and relaxed-nonnegativity conditions, and practical step-size control formulas that preserve stability while adapting to solution non-smoothness. The results demonstrate that adaptive splitting can faithfully reflect solution behavior in challenging parabolic-hyperbolic settings, offering a path toward efficient, physically faithful MHD simulations.
Abstract
We construct splitting methods suitable for the solution of the equations of magnetohydrodynamics (MHD). Due to the physical significance of the involved operators, splittings into three or even four operators with positive coefficients are appropriate for a physically correct and efficient solution of the equations. To efficiently obtain an accurate solution approximation, adaptive choice of the time-steps is important particularly in the light of the unsmooth dynamics of the system. Thus, we construct new method coefficients in conjunction with associated error estimators by optimizing the leading local error term. As a proof of concept, we demonstrate that adaptive splitting faithfully reflects the solution behavior also in the presence of a shock-like behavior for the viscous Burgers equation, which serves as a simplified model problem displaying several features of the Navier-Stokes equation for incompressible flow.