JOURNAL OF COMPUTATIONAL PHYSICS 133, 279–288 (1997) ARTICLE NO. CP975653 Asymptotically Stable Fourth-Order Accurate Schemes for the Diffusion Equation on Complex Shapes1

while constraining an energy norm of the error to be tem-porally bounded for all t. 0 by a constant proportionalAn algorithm which solves the multidimensional diffusion equa-tion 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 multidimen-matrix whose symmetric part is negative definite. The differentiation sional algorithm, even for irregular shapes containingmatrix 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 dimen-definitions, 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. Q 1997 method, in contradistinction to ‘‘standard’ ’ methods for a Academic Press Cartesian grid on irregular shapes. We consider the problemRecently there has been renewed interest in finite-differ