A meshfree framework for ideal magnetohydrodynamics

Abstract

A meshfree framework for numerical simulations of magnetohydrodynamic flows is proposed. The framework is based on the Least-Squares-Based Upwind Finite Difference method (LSFD-U) and is capable of handling arbitrary point distributions. The approach is based on the least-squares method of error minimisation to compute the inviscid flux derivatives. The flux derivatives at every point require the fluxes at the point as well as those in fictitious interfaces in a neighbourhood around it. The fluxes at the fictitious interfaces are computed by using any numerical flux formula of interest that incorporates upwinding and consequently a global rather than the one-sided stencil may be chosen for computation. The meshfree framework is implemented through AUSM scheme, using a first-order accurate spatial and temporal discretisation. Studies on the one-dimensional MHD shock tube problems demonstrate the efficacy of the algorithm. The effect of non-uniform point distributions on the performance of the meshfree framework particularly with regard to conservation has also been studied. On the non-uniform grid the solutions of meshfree framework are not in agreement with the solutions of the finite volume framework unlike on uniform meshes. The finite volume formulation is known to be conservative and therefore it is very clear that meshfree formulation has conservation issues particularly as the grid becomes more non-uniform. It is therefore desirable to have a formulation for meshfree framework that preserves conservation at a discrete level like the finite volume method. Attempts have been made to develop a conservative meshfree framework using weighted least-squares technique for a particular case of one-dimensional non-uniform point distribution.