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DOI: 10.1615/AnnualRevHeatTransfer.v1.50
pages 47-149

David J. Evans
Department of Computer Studies, Loughborough University of Technology, Loughborough, Leicestershire, UK


In this survey paper, Part 1 is concerned with new explicit methods for the finite difference solution of a parabolic partial differential equation in 1 space dimension. The new methods use stable asymmetric approximations to the partial differential equation which when coupled in groups of 2 adjacent points on the grid result in implicit equations which can be easily converted to explicit form which in turn offer many advantages. By judicious use of alternating this strategy on the grid points of the domain results in an algorithm which possesses unconditional stability. The merit of this approach results in more accurate solutions because of truncation error cancellations. The stability, consistency, convergence and truncation error of the new method is discussed and the results of numerical experiments presented.
Similarly, Part II briefly surveys existing methods and then an explicit finite difference approximation procedure which is unconditionally stable for the solution of the two dimensional non-homogeneous diffusion equation is presented. This method possesses the advantages of the implicit methods, i.e. no severe limitation on the size of the time increment. Also it has the simplicity of the explicit methods and employs the same "marching" type technique of solution. Results obtained by this method for several different problems were compared with the exact solution and agreed closely with those obtained by other finite-difference methods.

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