S. Abarbanel et A. Ditkowski, ASYMPTOTICALLY STABLE 4TH-ORDER ACCURATE SCHEMES FOR THE DIFFUSION EQUATION ON COMPLEX SHAPES, Journal of computational physics, 133(2), 1997, pp. 279-288
An algorithm which solves the multidimensional diffusion equation on c
omplex shapes to fourth-order accuracy and is asymptotically stable in
time is presented. This bounded-error result is achieved by construct
ing, on a rectangular grid, a differentiation matrix whose symmetric p
art is negative definite. The differentiation matrix accounts for the
Dirichlet boundary condition by imposing penalty-like terms. Numerical
examples in 2-D show that the method is effective even where standard
schemes, stable by traditional definitions, fail. The ability of the
paradigm to be applied to arbitrary geometric domains is an important
feature of the algorithm. (C) 1997 Academic Press.