PDE-W-methods for parabolic problems with mixed derivatives

The present work considers the numerical solution of differential equations that are obtained by space discretization (method of lines) of parabolic evolution equations. Main emphasis is put on the presence of mixed derivatives in the elliptic operator. An extension of the alternating-direction-implicit (ADI) approach to this situation is presented. Our stability analysis is based on a scalar test equation that is relevant to the considered class of problems. The novel treatment of mixed derivatives is implemented in third-order W-methods. Numerical experiments and comparisons with standard methods show the efficiency of the new approach. An extension of our treatment of mixed derivatives to 3D and higher dimensional problems is outlined at the end of the article.