A monotone finite volume scheme for advection–diffusion equations on distorted meshes

Abstract

SUMMARY A new monotone finite volume method with second‐order accuracy is presented for the steady‐state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes that guarantee the positivity of the numerical solution. The approximation of the diffusive flux is based on nonlinear two‐point approximation, and the approximation of the advective flux is based on the second‐order upwind method with proper slope limiter. The second‐order convergence rate for concentration and the monotonicity of the nonlinear finite volume method are verified with numerical experiments. Copyright © 2011 John Wiley & Sons, Ltd.

A Second-Order Positivity-Preserving Finite Volume Scheme for Diffusion Equations on General Meshes

“…The first one, originally suggested in [7] and named LPEW2 therein, has an explicit formula for interpolation weights. Recently, LPEW2 has been used by some authors [21,18]. The second algorithm LSW was suggested in [2] through a linear least squares approximation approach.…”

A new positive finite volume scheme for two‐dimensional convection‐diffusion equation

“…For discretization of the convective flux, some distinctive features have to be considered such as upwind characteristics. As we know, the gradient reconstruction method is one of the most popular methods. This method can keep the property of upwind.…”

A finite volume scheme preserving extremum principle for convection–diffusion equations on polygonal meshes

“…Copyright 617 so the nonnegativeness should be kept for their numerical approximations as well. Based on the idea of a two-point flux stencil, a few nonlinear FV schemes are developed further in a number of papers (see [31][32][33][34][35] and references therein), which guarantee monotonicity on general meshes for general tensor coefficients. In fact, the streamline upwind Petrov-Galerkin method is neither monotone nor monotonicity preserving.…”

“…And some algorithms based on slope limiters are proposed in [29] to preserve the monotonicity.A new research direction is pioneered by Le Potier [30], which uses a two-point flux stencil to construct a nonlinear FV scheme for highly anisotropic diffusion operators on unstructured triangular meshes, but the scheme is monotone only for parabolic equations and sufficiently small time steps. Based on the idea of a two-point flux stencil, a few nonlinear FV schemes are developed further in a number of papers (see [31][32][33][34][35] and references therein), which guarantee monotonicity on general meshes for general tensor coefficients.In [7], a nonlinear diamond scheme on unstructured meshes of d -simplices for convectiondiffusion problems is proposed, in which the face gradient is reformulated as a nonlinear average of the one-side gradients by suitably designing solution-dependent weights. A nonlinear FV scheme satisfying extremum principle for diffusion operators on triangular cells is presented in [36].…”

“…A new research direction is pioneered by Le Potier [30], which uses a two-point flux stencil to construct a nonlinear FV scheme for highly anisotropic diffusion operators on unstructured triangular meshes, but the scheme is monotone only for parabolic equations and sufficiently small time steps. Based on the idea of a two-point flux stencil, a few nonlinear FV schemes are developed further in a number of papers (see [31][32][33][34][35] and references therein), which guarantee monotonicity on general meshes for general tensor coefficients.…”