Application of least-squares meshfree method to a depth-averaged model of granular flow
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In recent years, more and more interest has been given to the study of geophysical flows: some softwares, such as TITAN2D, have been developed to predict their behaviour. While this methodology based on finite volume approximation has produced good simulations of a number of flows, many issues still remain open. In this work, we are going to use a depth-averaged model of granular flow proposed by Iverson and Denlinger, and a Least-Squares Meshfree Method (LSMM) approximation. Implementation of this method and derivation of the equations are also detailed, in particular for the simple problem of a flow along an inclined plane. Moreover several approximations in time has been studied: backward finite difference, space-time least-squares and forward finite-difference. The system of non-linear hyperbolic equations which has to be solved leads to oscillations of the approximate solution. The best results have been obtained with a backward finite difference scheme. In using some characteristics of the LSMM, we have been able to control temporarily the solution at the flow boundary where oscillations were initiated. Thus, an efficient boundary tracking method has been implemented. A stabilization method, based on the introduction of an artificial diffusion-type term in the least-squares minimization, has controlled on a longer time the oscillations.