We consider a scalar conservation law with zero-flux boundary conditions imposed on the boundary of a rectangular multidimensional domain. We study monotone schemes applied to this problem. For the Godunov version of the scheme, we simply set the boundary flux equal to zero. For other monotone schemes, we additionally apply a simple modification to the numerical flux. We show that the approximate solutions produced by these schemes converge to the unique entropy solution, in the sense of [7], of the conservation law. Our convergence result relies on a BV bound on the approximate numerical solution. In addition, we show that a certain functional that is closely related to the total variation is nonincreasing from one time level to the next. We extend our scheme to handle degenerate convection-diffusion equations and for the one-dimensional case we prove convergence to the unique entropy solution.

Nonnegative directional splittings of anisotropic diffusion operators in the divergence form are investigated. Conditions are established for nonnegative directional splittings to hold in a neighborhood of an arbitrary interior point. The result is used to construct monotone finite difference schemes for the boundary value problem of anisotropic diffusion operators. It is shown that such a monotone scheme can be constructed if the underlying diffusion matrix is continuous on the closure of the physical domain and symmetric and uniformly positive definite on the domain, the mesh spacing is sufficiently small, and the size of finite difference stencil is sufficiently large. An upper bound for the stencil size is obtained, which is determined completely by the diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the larger stencil is required. An exception is the situation with a strictly diagonally dominant diffusion matrix where a three-by-three stencil is sufficient for the construction of a monotone finite difference scheme. Numerical examples are presented to illustrate the theoretical findings.

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whoseleaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate thecomputational efficiency together with the convergence properties.

First–order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a (2K+1)-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.