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

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 time-dependent 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 h-refinement are presented, which cover many intriguing aspects typifying nonlinear magnetofluid dynamics, with higher accuracy than often seen in similar publications.