Construction of adaptive exponential multi-operator splitting methods

TL;DR

This paper develops adaptive exponential multi-operator splitting methods tailored for magnetohydrodynamics equations, incorporating positive coefficient splittings and error estimators to handle complex, unsmooth dynamics effectively.

Contribution

It introduces new splitting method coefficients with error estimators optimized for accurate, adaptive solutions of MHD equations, including shock-like behaviors.

Findings

Adaptive splitting methods accurately capture shock-like dynamics.

New coefficients improve solution efficiency and physical consistency.

Method demonstrates effectiveness on simplified Navier-Stokes-like models.

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.