This paper deals with the numerical solution of the Heston partial
differential equation (PDE) that plays an important role in financial option
pricing theory, Heston (1993). A feature of this time-dependent, twodimensional
convection-diffusion-reaction equation is the presence of a mixed
spatial-derivative term, which stems from the correlation between the two underlying
stochastic processes for the asset price and its variance.
Semi-discretization of the Heston PDE, using finite difference schemes on
non-uniform grids, gives rise to large systems of stiff ordinary differential equations.
For the effective numerical solution of these systems, standard implicit
time-stepping methods are often not suitable anymore, and tailored timediscretization
methods are required. In the present paper, we investigate four
splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas
scheme, the Craig-Sneyd scheme, the Modified Craig-Sneyd scheme, and the
Hundsdorfer