Godunov-Type Numerical Methods for a Model of Granular Flow on Open Tables with Walls (Communications in Computational Physics, 2018)
We propose and analyse finite volume Godunov type methods based on discontinuousflux for a 2×2 system of non-linear partial differential equations proposedby Hadeler and Kuttler to model the dynamics of growing sandpiles generated by avertical source on a flat bounded rectangular table. The problem considered here isthe so-called partially open table problem where sand is blocked by a wall (of infiniteheight) on some part of the boundary of the table. The novelty here is the correspondingmodification of boundary conditions for the standing and the rolling layers andgeneralization of the techniques of the well-balancedness proposed in [1]. Presenceof walls may lead to unbounded or discontinuous surface flow density at equilibriumresulting in solutions with singularities propagating from the extreme points of thewalls. A scheme has been proposed to approximate efficiently the Hamiltonians withthe coefficients which can be unbounded and discontinuous. Numerical experimentsare presented to illustrate that the proposed schemes detect these singularities in theequilibrium solutions efficiently and comparisons are made with the previously studiedfinite difference and Semi-Lagrangian approaches by Finzi Vita et al.