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Numerical analysis of three‐time‐level finite difference schemes for unsteady diffusion–convection problems
Author(s) -
Rigal Alain
Publication year - 1990
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620300207
Subject(s) - mathematics , numerical analysis , convection–diffusion equation , numerical diffusion , diffusion , finite difference method , numerical stability , finite difference , stability (learning theory) , partial differential equation , reynolds number , dissipation , mathematical analysis , computer science , mechanics , physics , turbulence , machine learning , thermodynamics
For the linear diffusion–convection model problem, this paper develops the numerical analysis of two classes of three‐time‐level second order finite difference schemes: weighed schemes similar to two‐step schemes for differential equations and LF schemes based on the classical conservative approximation of the wave equation. As in our paper devoted to two‐level schemes, Fourier Analysis is an essential tool which yields several key properties (stability, dissipation, numerical diffusion and dispersion). It also permits the analysis of parasitic solutions characteristic of three‐level schemes and their control in terms of the starting scheme. The analysis is completed by giving results on the accuracy, positivity and parabolicity of the numerical schemes. The main conclusions are the expected superiority of these schemes over two‐level schemes when the diffusion phenomenon is not negligible (cell Reynolds number less than 1) and the very limited possibilities of obtaining satisfactory numerical results (even with upstream differencing) when convection is strongly predominating.