Analysis of explicit difference methods for a diffusion‐convection equation

Abstract

SUMMARYWe consider the numerical solution of a model one-dimensional diff usion-convection equation by a variety of explicit finite difference methods including conventional central and upwind replacements of the convection terms. We discuss commonly observed phenomena such as instability, unwanted oscillations in the numerical solution, and numerical diffusion and we present an analysis of these effects by simple mathematical techniques.

Explicit finite-difference solution of two-dimensional solute transport with periodic flow in homogenous porous media

“…Notable numerical techniques are finite difference methods (FDM), finite element methods (FEM) and finite volume methods (FVM) -naturally, all three discretize governing equations and initial and boundary conditions. For ADE with both constant and variable coefficients, Ahmed (2012) proposed an FDM discretization scheme that combined the Siemieniuch-Gradwell's approximation (Siemieniuch and Gladwell, 1978) for time and Dehghan's approximation (Dehghan, 2004) for the spatial variable. Savović and Djordjevich (2012) solved the ADE with variable coefficients in semi-infinite domain by an explicit formulation of the finite difference method (EFDM).…”

On the Convective and Absolute Nature of Instabilities in Finite Difference Numerical Simulations of Open Flows

“…The curve which separates the stable from the unstable regions is the classical result of the von Neumann stability analysis (see, for example, [6]). In this case it is found that the absolute instability boundary, found by a local analysis, exactly coincides with the global instability boundary, obtained with the spectral radius criterion, when Dirichlet boundary conditions are enforced [6,12]. The absolute nature of the instability allows an amplified energy radiation from the downstream boundary into the computational domain and thus a global instability.…”

A compact unconditionally stable finite‐difference method for transient one‐dimensional advection‐diffusion

“…Finally, equation (24) of this NEW second-order method has an MEPDE of the form so, for A x 4 1, the dominant term in the truncation error is second order in A x and is proportional to q3(c, s). For equation (24), qj(c,s) takes the form…”

“…Note that this procedure does not require a knowledge of (4), which permits the scheme to be implemented without the requirement that values at the downstream boundary x = X must be known in addition to ? (O, 1").…”

“…( X , t ) = gR(f), O < f < T (3) (4) The scalar function ? (x, t) may represent, for example, temperature' or concentration of a passive solute' at position x and time t in a fluid moving with speed u and subject to diffusion governed by the coefficient a .…”