The stability of explicit Euler time‐integration for certain finite difference approximations of the multi‐dimensional advection–diffusion equation

Abstract

SUMMARYA comprehensive study is presented regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection-difision equation. One-dimensional results are obtained using both the matrix method (for several boundary conditions) and the classical von Neumann method of stability analysis and arguments presented showing that the latter is generally to be preferred, regardless of the type of boundary conditions. The less-well-known Godunov-Ryabenkii theory is also applied for a particular (Robin) boundary condition. After verifying portions of the one-dimensional theory with some numerical results, the stabilities of the two-and three-dimensional equations are addressed using the von Neumann method and results presented in the form of a new stability theorem. Extension of a useful scheme from one dimension, where the pure advection limit is known variously as kith's method or a Lax-Wendroff method, to many dimensions via finite elements is also addressed and some stability results presented.KEY WORDS Stability Advectiondfision von Neumann method Matrix method Explicit Euler

von Neumann stability conditions for the convection-diffusion eqation

“…With a = b = \, Theorem 3.1 immediately gives sufficient stability conditions. In the case K = 1 (second-order central scheme) these conditions are shown to be also necessary in Wesseling (1995), and identical to those obtained in Hirt (1968), Morton (1971), Hindmarsh et al (1984). The corresponding restrictions on r are easily found from (3.15) and (3.18) by substitution of the relevant values for a and b.…”

Numerical error in groundwater flow and solute transport simulation

“…[20] The numerical stability range of a solution technique can sometimes be determined analytically; for complicated techniques the stability range can be more difficult to establish [Hindmarsh et al, 1984;Noye, 1991]. Voss [1984] suggests a provisional stability constraint of Áx 4b, where b is the longitudinal dispersivity.…”

On the odd-even hopscotch scheme for the numerical integration of time-dependent partial differential equations

“…Hence one might also consider the use of simple one-sided spatial differencing. With regard to the convection-diffusion equation (1.1) we note that diffusion terms of the type ' [9], are not allowed because of the cross-derivatives.…”

“…Via this equivalence we derive the explicit expression of the critical time step for von Neumann stability for problem (1.1). This expression is easily found by applying a useful theorem due to Hindmarsh, Gresho and Griffiths [9]. This theorem plays an important role in their stability analysis of the forward Euler-central difference scheme.…”

“…Firstly, via the equivalence to the combined leapfrog-Du Fort-Frankel method we derive the explicit expression of the critical time step for von Neumann stability for a class of multi-dimensional convection-diffusion equations. This expression can be derived directly by applying a useful stability theorem due to Hindmarsh, Gresho and Griffiths [9]. The interesting thing on the critical time step is that it is independent of the diffusion parameter and yet smaller than the critical time step for zero diffusion, but only in the multi-dimensional case.…”