PDE-W-methods for parabolic problems with mixed derivatives
Severiano González-Pinto, Ernst Hairer, Domingo Hernández-Abreu
and Soledad Pérez-Rodríguez
Abstract. 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 $3$rd 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.
Key Words. Parabolic problems, mixed derivatives, W-methods, stability analysis.