A Note on Finite Differencing of the Advection-Diffusion Equation

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  • 1 Environmental Models Branch, Naval Ocean Research and Development Activity, NSTL Station, MS 39529
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Abstract

The criteria advanced by Fromm (1964) and Roache (1976) as necessary and sufficient conditions for numerical stability of the forward-in-time, centered-in-space finite-difference treatment of the advection-diffusion equation are shown to be sufficient but not necessary conditions for stability. The correct necessary and sufficient conditions for stability are derived and discussed. Since these conditions are much less restrictive than those advanced by the earlier authors, they will allow use of this scheme in a wider class of problems. Results from a long-term and stable integration using parameters which are characteristic of large-scale ocean models and which grossly violate the earlier stability criteria are presented and compared to an exact analytical solution.

Abstract

The criteria advanced by Fromm (1964) and Roache (1976) as necessary and sufficient conditions for numerical stability of the forward-in-time, centered-in-space finite-difference treatment of the advection-diffusion equation are shown to be sufficient but not necessary conditions for stability. The correct necessary and sufficient conditions for stability are derived and discussed. Since these conditions are much less restrictive than those advanced by the earlier authors, they will allow use of this scheme in a wider class of problems. Results from a long-term and stable integration using parameters which are characteristic of large-scale ocean models and which grossly violate the earlier stability criteria are presented and compared to an exact analytical solution.

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