This paper presents a comparative study on locally mass-conservative numerical methods for Darcy’s flows. The classical mixed finite element method (MFEM) is compared with the newly developed discontinuous finite volume method (DFVM) with and without weak over-penalization (WOP). These numerical methods are tested on three representative problems in porous media flows. In particular, locality, ...

In [19, 20, 22], we constructed uniformly high order accurate discontinuous Galerkin (DG) which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. The technique also applies to high order accurate finite volume schemes. In this paper, we show an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (E...

Abstract. We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using so-called complementary volumes to a finite element triangulation. The scheme gives solution in ...

Abstract. We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. I...

Time-domain finite difference (FD) and digital waveguide mesh (DWM) methods have seen extensive exploration as techniques for physical modelling sound synthesis and artificial reverberation. Various formulations of these methods have been unified under the FD framework, but many discrete boundary models important in room acoustics applications have not been. In this paper, the finite volume (FV...

Finite volume moving mesh methods are developed for unsteady one-dimensional partial diierential equations. The methods can be viewed as conservative discretisations of the Lagrangian form of the original diier-ential equations. Grid points are moved using the idea of mesh equidistri-bution. The coupled unsteady diierential equations and grid movement equations are solved using a robust and eec...

Based on general partitions of unity and standard numerical flux functions, a class of mesh-free methods for conservation laws is derived. A Lax-Wendroff type consistency analysis is carried out for the general case of moving partition functions. The analysis leads to a set of conditions which are checked for the finite volume particle method FVPM. As a by-product, classical finite volume schem...

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite e...

A main issue in nonstationary, compressible magnetohydrodynamic (MHD) simulations is controlling the divergence of the magnetic flux. This paper presents a general procedure showing how to modify the intercell fluxes in a conservative MHD finite volume code such that the scheme becomes locally divergence preserving. That is, a certain discrete divergence operator vanishes exactly during the ent...

A simple and accurate projection finite volume method is developed for solving shallow water equations in two space dimensions. The proposed approach belongs to the class of fractional-step procedures where the numerical fluxes are reconstructed using the method of characteristics, while an Eulerian method is used to discretize the conservation equations in a finite volume framework. The method...