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

• A scheme is proposed for variable coefficient parabolic PDE with mixed derivative.
• It is a compact, stable, fourth order scheme with good dispersive property.
• The scheme is suitable for computing incompressible flow in arbitrary domains.
• The scheme is proficient in conjunction with numerically generated grids.
• It is a first for non-conservative form of the incompressible N-S equations in curvilinear grid.

A fourth order compact numerical scheme for variable coefficient parabolic problems with mixed derivatives is outlined. The finite difference scheme, presented here shows good wave resolution property and is stable. Implicit time discretization endows second order temporal accurateness to the scheme. Compact scheme for generalized parabolic 2D convection-diffusion equations is not available in the literature and this work addresses the same. The method is also suitable for computing incompressible flow in arbitrary domains. In this work we have successfully used the method to tackle flows, governed by the incompressible 2D unsteady Navier-Stokes (N-S) equations, in regions beyond rectangular. Further the proposed compact scheme has been found to be proficient in conjunction with numerically generated grids.