A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics

Abstract

In this paper we describe a one-dimensional adaptive moving mesh method and its application to hyperbolic conservation laws from magnetohydrodynamics (MHD). The method is robust, because it employs automatic control of mesh adaptation when a new model is considered, without manually-set parameters. Adaptive meshes are a common tool for increasing the accuracy and reducing computational costs when solving time-dependent partial differential equations (PDEs). Mesh points are moved towards locations where they are needed the most. To obtain a timedependent adaptive mesh, monitor functions are used to automatically ‘monitor’ the importance of the various parts of the domain, by assigning a ‘weight’-value to each location. Based on the equidistribution principle, all mesh points are distributed according to their assigned weights. We use a sophisticated monitor function that tracks both small, local phenomena as well as large shocks in the same solution. The combination of the moving mesh method and a high-resolution finite volume solver for hyperbolic PDEs yields a serious gain in accuracy at relatively no extra costs. The results of several numerical experiments —including comparisons with hrefinement— are presented, which cover many intriguing aspects typifying nonlinear magnetofluid dynamics, with higher accuracy than often seen in similar publications. - PACS: 02.70.Bf, 52.30.Cv, 52.35.Bj, 52.35.Tc, 52.65.Kj 2000 MSC: 35L60, 35L65, 65M50, 76L05, 76M12, 76W05