  We present a high-order compact finite difference approach for a class of
parabolic partial differential equations with time and space dependent
coefficients as well as with mixed second-order derivative terms in $n$ spatial
dimensions. Problems of this type arise frequently in computational fluid
dynamics and computational finance. We derive general conditions on the
coefficients which allow us to obtain a high-order compact scheme which is
fourth-order accurate in space and second-order accurate in time. Moreover, we
perform a thorough von Neumann stability analysis of the Cauchy problem in two
and three spatial dimensions for vanishing mixed derivative terms, and also
give partial results for the general case. The results suggest unconditional
stability of the scheme. As an application example we consider the pricing of
European Power Put Options in the multidimensional Black-Scholes model for two
and three underlying assets. Due to the low regularity of typical initial
conditions we employ the smoothing operators of Kreiss et al. to ensure
high-order convergence of the approximations of the smoothed problem to the
true solution.
