We propose and analyse finite volume Godunov type methods based on discontinuous flux for a 2×2 system of non-linear partial differential equations proposed by Hadeler and Kuttler to model the dynamics of growing sandpiles generated by a vertical source on a flat bounded rectangular table. The problem considered here is the so-called partially open table problem where sand is blocked by a wall (of infinite height) on some part of the boundary of the table. The novelty here is the corresponding modification of boundary conditions for the standing and the rolling layers and generalization of the techniques of the well-balancedness proposed in [1]. Presence of walls may lead to unbounded or discontinuous surface flow density at equilibrium resulting in solutions with singularities propagating from the extreme points of the walls. A scheme has been proposed to approximate efficiently the Hamiltonians with the coefficients which can be unbounded and discontinuous. Numerical experiments are presented to illustrate that the proposed schemes detect these singularities in the equilibrium solutions efficiently and comparisons are made with the previously studied finite difference and Semi-Lagrangian approaches by Finzi Vita et al.

For scalar conservation laws in one space dimension with a flux function discontinuous inspace, there exist infinitely many classes of solutions which are L1 contractive.Each class is characterized by a connection (A,B) which determines the interface entropy. Forsolutions corresponding to a connection (A,B), there exists convergent numerical schemesbased on Godunov or Engquist−Osher schemes. The natural question is how to obtain schemes,corresponding to computationally less expensive monotone schemes like Lax−Friedrichs etc., usedwidely in applications. In this paper we completely answer this question for more general(A,B)stable monotone schemes using a novel construction of interface flux function. Then fromthe singular mapping technique of Temple and chain estimate of Adimurthi and Gowda, weprove the convergence of the schemes.

Long time simulations of transport equations raise computational challenges since theyrequire both a large domain of calculation and sufficient accuracy. It is thereforeadvantageous, in terms of computational costs, to use a time varying adaptive mesh, withsmall cells in the region of interest and coarser cells where the solution is smooth.Biological models involving cell dynamics fall for instance within this framework and areoften non conservative to account for cell division. In that case the thresholdcontrolling the spatial adaptivity may have to be time-dependent in order to keep up withthe progression of the solution. In this article we tackle the difficulties arising whenapplying a Multiresolution method to a transport equation with discontinuous fluxesmodeling localized mitosis. The analysis of the numerical method is performed on asimplified model and numerical scheme. An original threshold strategy is proposed andvalidated thanks to extensive numerical tests. It is then applied to a biological model inboth cases of distributed and localized mitosis.