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.

We propose an all regime Lagrange-Projection like numerical scheme for the gas dynamics equations. By all regime, we mean that the numerical scheme is able to compute accurate approximate solutions with an under-resolved discretization with respect to the Mach number M, i.e. such that the ratio between the Mach number M and the mesh size or the time step is small with respect to 1. The key idea is to decouple acoustic and transport phenomenon and then alter the numerical flux in the acoustic approximation to obtain a uniform truncation error in term of M. This modified scheme is conservative and endowed with good stability properties with respect to the positivity of the density and the internal energy. A discrete entropy inequality under a condition on the modification is obtained thanks to a reinterpretation of the modified scheme in the Harten Lax and van Leer formalism. A natural extension to multi-dimensional problems discretized over unstructured mesh is proposed. Then a simple and efficient semi implicit scheme is also proposed. The resulting scheme is stable under a CFL condition driven by the (slow) material waves and not by the (fast) acoustic waves and so verifies the all regime property. Numerical evidences are proposed and show the ability of the scheme to deal with tests where the flow regime may vary from low to high Mach values.

Inspired by the growing use of non linear discretization techniques for the lineardiffusion equation in industrial codes, we construct and analyze various explicit nonlinear finite volume schemes for the heat equation in dimension one. These schemes areinspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I348 (2010) 691–695]. They preserve the maximum principle and admita finite volume formulation. We provide a original functional setting for the analysis ofconvergence of such methods. In particular we show that the fourth discrete derivative isbounded in quadratic norm. Finally we construct, analyze and test a new explicit nonlinear maximum preserving scheme with third order convergence: it is optimal on numericaltests.

We propose an a-posteriori error/smoothness indicator for standard semi-discrete finite volume schemes for systems of conservation laws, based on the numerical production of entropy. This idea extends previous work by the first author limited to central finite volume schemes on staggered grids. We prove that the indicator converges to zero with the same rate of the error of the underlying numerical scheme on smooth flows under grid refinement. We construct and test an adaptive scheme for systems of equations in which the mesh is driven by the entropy indicator. The adaptive scheme uses a single nonuniform grid with a variable timestep. We show how to implement a second order scheme on such a space-time non uniform grid, preserving accuracy and conservation properties. We also give an example of a p-adaptive strategy.

We consider constraint preserving multidimensional evolution equations. A prototypical example is provided by the magnetic induction equation of plasma physics. The constraint of interest is the divergence of the magnetic field. We design finite volume schemes which approximate these equations in a stable manner and preserve a discrete version of the constraint. The schemes are based on reformulating standard edge centered finite volume fluxes in terms of vertex centered potentials. The potential-based approach provides a general framework for faithful discretizations of constraint transport and we apply it to both divergence preserving as well as curl preserving equations. We present benchmark numerical tests which confirm that our potential-based schemes achieve high resolution, while being constraint preserving.

In this paper we consider two commonly used classes of finite volume weighted essentially non-oscillatory (WENO) schemes in two dimensional Cartesian meshes. We compare them in terms of accuracy, performance for smooth and shocked solutions, and efficiency in CPU timing. For linear systems both schemes are high order accurate, however for nonlinear systems, analysis and numerical simulation results verify that one of them (Class A) is only second order accurate, while the other (Class B) is high order accurate. The WENO scheme in Class A is easier to implement and costs less than that in Class B. Numerical experiments indicate that the resolution for shocked problems is often comparable for schemes in both classes for the same building blocks and meshes, despite of the difference in their formal order of accuracy. The results in this paper may give some guidance in the application of high order finite volume schemes for simulating shocked flows.

We study a depth-averaged model of gravity-driven flows made ofsolid grains and fluid, moving over variable basal surface. In particular, we are interested in applicationsto geophysical flows such as avalanches and debris flows,which typically contain both solid material and interstitial fluid.The model system consists of mass and momentum balance equations for thesolid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns,and by interphase drag source terms. The system is hyperbolic at leastwhen the difference between solid and fluid velocities is sufficiently small.We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing oftopography source terms is obtained via a technique that includesthese contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problemsof perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

We construct finite volume schemes, on unstructured and irregular grids andin any space dimension, for non-linear elliptic equations of the p-Laplacian kind: -div(|∇u|p-2 ∇u) = ƒ(with 1 < p < ∞). We prove the existence and uniqueness of the approximate solutions,as well as their strong convergence towards the solution of the PDE.The outcome of some numerical tests are also provided.

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t + \nabla \cdot ( {\bf u}f(c)) - \nabla \cdot (D \nabla c) + \lambda c = 0$ .The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L 1-norm,independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability of the theoretical results.