An adaptive spline wavelet ADI (SW-ADI) method for two-dimensional reaction-diffusion equations

We study a spline wavelet alternative direction implicit (SW-ADI) algorithm for solving two-dimensional reaction diffusion equations. This algorithm is based on a collocation method for PDEs with a specially designed spline wavelet for the Sobolev space H-2(I) on a closed interval I. By using the tensor product nature of adaptive wavelet meshes, we propose a SW-ADI method for two-dimensional problems. The proposed SW-ADI method is an efficient time-dependent adaptive method with second-order accuracy for solutions with localized phenomena, such as in flame propagations. Issues like new boundary wavelets for more accurate boundary conditions, adaptive strategy for two-dimensional meshes, data structure and storage and implementation details, and numerical results will be discussed. (C) 1998 Academic Press.