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In this article, we have developed a higher order compact numerical methodfor variable coefficient parabolic problems with mixed derivatives. The finitedifference scheme, presented here for two-dimensional domains, is based onfourth order spatial discretization. The time discretization has been carriedout using using second order Crank-Nicolson. The present scheme shows gooddispersion relation preserving property and has been thoroughly investigatedfor stability. The discrete Fourier analysis shows that the method isunconditionally stable. The fact that the method has been particularlydeveloped for parabolic equations with mixed derivatives makes it suitable forsolving incompressible Navier-Stokes (N-S) equations in irregular domains. Toverify the proposed method, several problems with exact and benchmark solutionshas been investigated. The proposed compact discretization has been extended totackle flows of varying complexities governed by the two-dimensional unsteadyN-S equations in domain beyond rectangular. The results show good agreement forall the problems considered.

Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives