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

TL;DR

This paper develops a third-order Runge-Kutta discontinuous Galerkin method to numerically solve the Savage-Hutter equations, effectively modeling granular avalanches with shock waves and vacuum fronts on inclined surfaces.

Contribution

It introduces a high-order RKDG scheme with a TVD slope limiter and well-balanced treatment for the Savage-Hutter equations, improving accuracy and stability in granular flow simulations.

Findings

The scheme accurately captures avalanche dynamics with various friction and slope angles.

Numerical results demonstrate the method's effectiveness in modeling granular avalanches.

The approach maintains the well-balanced property for the reposing state.

Abstract

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.