Multiparameter families of difference approximations for the first initial boundary value problem for the heat equation in an arbitrary region

✍ Scribed by J. H. Giese; H. M. Khalil; D. L. Ulery

• Publisher: Springer
• Year: 1978
• Tongue: English
• Weight: 738 KB
• Volume: 12
• Category: Article
• ISSN: 0022-0833
• DOI: 10.1007/bf00043213

✦ Synopsis

A symbolic technique is developed to automatically generate consistent multiparameter families of difference approximations to the heat equation with Dirichlet boundary conditions in arbitrary regions. A stencil of nonuniform step size, conformable along the spatial axes via spatial displacement parameters, is devised to handle the problem of irregular boundaries. Using this stencil as a basic building block, multiparameter families of difference schemes applicable without modification both in the interior and along the boundaries of arbitrary regions, are algorithmically generated. The technique is demonstrated in detail for the one-and two-dimensional heat operator. Necessary and sufficient conditions for the stability of these families are given in terms of their parameters. All existing six-and ten-point two-level schemes for the one-and two-dimensional cases are shown to form subclasses of these families.