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AMF-type W-methods for parabolic problems with mixed derivatives

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Severiano González-Pinto, Ernst Hairer, Domingo Hernández-Abreu and Soledad Pérez-Rodríguez,

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Abstract. The time integration of differential equations obtained by the space discretization via Finite Differences of evolution parabolic PDEs with mixed derivatives in the elliptic operator is considered (MOL approach). W-methods (Rosenbrock-type methods) are combined with the Approximate Matrix Factorization technique (AMF), which is considered in alternating direction implicit (ADI) sense. The focus of the paper is a stability analysis, which is based on a scalar test equation that is relevant for the class of problems when periodic or homogeneous Dirichlet boundary conditions are considered. Unconditional stability, independent of the number of space dimensions $m$, is proved for a variety of AMF-type W-methods. Numerical experiments with linear parabolic problems in dimension m=3 and m=4, as well as with the Heston problem from financial option pricing are presented.

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Key Words. Parabolic PDEs, mixed derivatives, time integration, W-methods, Approximate Matrix Factorization, Alternating Direction Implicit schemes, unconditional stability, Heston model.

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