A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics

Pages: 161 - 178, Volume 8, Issue 1, January 2009

doi:10.3934/cpaa.2009.8.161       Abstract        Full Text (2474.7K)       Related Articles

Alexandre Caboussat - University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204 - 3008, United States (email)
Roland Glowinski - University of Houston, Department of Mathematics, 4800 Calhoun Rd, Houston, Texas 77204 - 3008, United States (email)

Abstract: An operator-splitting algorithm is presented for the solution of a partial differential equation arising in the modeling of deposition processes in sand mechanics. Sand piles evolution is modeled by an advection-diffusion equation, with a non-smooth diffusion operator that contains a point-wise constraint on the gradient of the solution. Piecewise linear finite elements are used for the discretization in space. The advection operator is treated with a stabilized SUPG finite element method. An augmented Lagrangian method is proposed for the discretization of the fast/slow non-smooth diffusion operator. A penalization approach, together with a Newton method, is used for the treatment of inequality constraints. Numerical results are presented for the simulation of sand piles on flat and non-flat surfaces, and for extensions to water flows.

Keywords:  Operator splitting, transport equation, non-smooth diffusion, augmented Lagrangian method, finite element approximation, point-wise constraints
Mathematics Subject Classification:  Primary: 65N30, 65M60, 65K10, 49S05; Secondary: 74S05

Received: May 2008;      Revised: August 2008;      Published: October 2008.