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Numerical Solution of the Savage-Hutter Equations for Granular Avalanche Flow using the Discontinuous Galerkin Method

The Savage-Hutter (SH) equations are a hyperbolic system of nonlinear partial differential equations describing the temporal evolution of the depth and depth averaged velocity for modelling the avalanche of a shallow layer of granular materials on an inclined surface. These equations admit the occurrence of shock waves and vacuum fronts as in the shallow-water equations while possessing the special reposing state of granular material. In this paper, we develop a third-order Runge-Kutta discontinuous Galerkin (RKDG) method for the numerical solution of the one-dimensional SH equations. We adopt a TVD slope limiter to suppress numerical oscillations near discontinuities. And we give numerical treatments for the avalanche front and for the bed friction to achieve the well-balanced reposing property of granular materials. Numerical results of the avalanche of cohesionless dry granular materials down an inclined and smoothly transitioned to horizontal plane under various internal and bed friction angles and slope angles are given to show the performance of the present numerical scheme.