AMFR-W-methods for parabolic problems with mixed derivates. Applications to the Heston model

Abstract

The combination of Approximate Matrix Factorization (AMF), W-methods and iterative refinement in the solution of linear systems leads to the definition of AMFR-W-methods. This method class provides stable and accurate time integrators for parabolic PDEs with mixed derivatives discretized in space by means of Finite Differences (or Finite Volumes) in an arbitrary number of spatial dimensions. When the coefficients of the PDE actually depend on the spatial variables, the approximation of the pure diffusion coefficients by its respective maximum value produces simplified AMFR-W-methods requiring only a reduced number of LU decompositions of banded matrices with small bandwidth. The new class of methods is shown to be unconditionally stable regardless of the spatial dimension on a linear test problem relevant for homogeneous or periodic boundary conditions. Furthermore, high orders of convergence in PDE sense are observed when homogeneous boundary conditions are assumed. For general Robin boundary conditions, a simple algorithm is provided to convert a PDE problem into one where such conditions are homogeneous. Numerical experiments with the new simplified AMFR-W-methods on a linear parabolic problem with variable coefficients and the Heston problem from financial option pricing are presented.