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We present a high-order compact finite difference approach for a class ofparabolic partial differential equations with time and space dependentcoefficients as well as with mixed second-order derivative terms in $n$ spatialdimensions. Problems of this type arise frequently in computational fluiddynamics and computational finance. We derive general conditions on thecoefficients which allow us to obtain a high-order compact scheme which isfourth-order accurate in space and second-order accurate in time. Moreover, weperform a thorough von Neumann stability analysis of the Cauchy problem in twoand three spatial dimensions for vanishing mixed derivative terms, and alsogive partial results for the general case. The results suggest unconditionalstability of the scheme. As an application example we consider the pricing ofEuropean Power Put Options in the multidimensional Black-Scholes model for twoand three underlying assets. Due to the low regularity of typical initialconditions we employ the smoothing operators of Kreiss et al. to ensurehigh-order convergence of the approximations of the smoothed problem to thetrue solution.

High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions