  In this article, we have developed a higher order compact numerical method
for variable coefficient parabolic problems with mixed derivatives. The finite
difference scheme, presented here for two-dimensional domains, is based on
fourth order spatial discretization. The time discretization has been carried
out using using second order Crank-Nicolson. The present scheme shows good
dispersion relation preserving property and has been thoroughly investigated
for stability. The discrete Fourier analysis shows that the method is
unconditionally stable. The fact that the method has been particularly
developed for parabolic equations with mixed derivatives makes it suitable for
solving incompressible Navier-Stokes (N-S) equations in irregular domains. To
verify the proposed method, several problems with exact and benchmark solutions
has been investigated. The proposed compact discretization has been extended to
tackle flows of varying complexities governed by the two-dimensional unsteady
N-S equations in domain beyond rectangular. The results show good agreement for
all the problems considered.
