Asymptotically Stable Fourth-Order Accurate Schemes for the Diffusion Equation on Complex Shapes

✍ Scribed by Saul Abarbanel; Adi Ditkowski

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 419 KB
• Volume: 133
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1997.5653

✦ Synopsis

while constraining an energy norm of the error to be temporally bounded for all t Ͼ 0 by a constant proportional An algorithm which solves the multidimensional diffusion equation on complex shapes to fourth-order accuracy and is asymptoti-to the truncation error.

cally stable in time is presented. This bounded-error result is

In Section 3 it is shown how the methodology developed achieved by constructing, on a rectangular grid, a differentiation in Section 2 is used as a building block for the multidimenmatrix whose symmetric part is negative definite. The differentiation sional algorithm, even for irregular shapes containing matrix accounts for the Dirichlet boundary condition by imposing ''holes''.

penalty-like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional Section 4 presents numerical results in two space dimendefinitions, fail. The ability of the paradigm to be applied to arbitrary sions illustrating the long-time temporal stability of the geometric domains is an important feature of the algorithm. ᮊ 1997 method, in contradistinction to ''standard'' methods for a

Academic Press

Cartesian grid on irregular shapes.