Stability properties of the Discontinuous Galerkin Material Point Method for hyperbolic problems in one and two space dimensions

In this paper, stability conditions are derived for the Discontinuous Galerkin Material Point Method (DGMPM) on the scalar linear advection equation for the sake of simplicity and without loss of generality for linear problems. The discrete systems resulting from the application of the DGMPM discretization in one and two space dimensions are first written. For these problems, a second-order Runge-Kutta and the forward Euler time discretizations are respectively considered. Moreover, the numerical fluxes are computed at cell faces by means of either the Donor-Cell Upwind or the Corner Transport Upwind methods for multidimensional problems. Second, the discrete scheme equations are derived assuming that all cells of a background grid contain at least one particle. Although a Cartesian grid is considered in two space dimensions, the results can be extended to regular grids. The von Neumann linear stability analysis then allows the computation of the critical Courant number for a given space discretization. Although the DGMPM is equivalent to the first-order finite volume method if one particle lies in each element, so that the Courant number can be set to unity, other distributions of particles may restrict the stability region of the scheme. The study of several configurations is then proposed.