A new compact difference scheme for second derivative in non-uniform grid expressed in self-adjoint form

A single-parameter family of self-adjoint compact difference (SACD) schemes is developed for discretizing the Laplacian operator in self-adjoint form. Developed implicit scheme is formally second-order accurate (with respect to truncation error) with a free parameter, @a which helps control the numerical properties in the spectral plane. The SACD scheme is analyzed in the spectral plane for its resolution properties for both periodic and non-periodic problems using the matrix spectral analysis [T.K. Sengupta, G. Ganeriwal, S. De, Analysis of central and upwind schemes, J. Comput. Phys. 192 (2) (2003) 677-694]. The major objective here is to identify the advantages of the new scheme over the traditional explicit second order CD2 scheme, in discretizing the Laplacian operator in self-adjoint form. This appears in Navier-Stokes equation and in other transport equations and boundary value problems (bvp) expressed in orthogonal co-ordinate systems, either in physical or in transformed plane. We also compare the developed method with the higher order compact schemes for non-uniform grids. To demonstrate the accuracy of SACD scheme we have tested it for: (i) bi-directional wave propagation problem, given by the second order wave equation and (ii) an elliptic bvp, as in the Stommel ocean model for the stream function. These examples help infer the properties of SACD scheme when solving different types of partial differential equations. Most importantly the effects of grid-stretching and choice of value of the free parameter (@a) are investigated here. We also compare the present implicit compact method with explicit compact method known as the higher order compact (HOC) method. Also, the practical applications of the SACD scheme are explored by solving the Navier-Stokes equation for the cases of: (a) a flow inside a lid-driven cavity and (b) the receptivity and instability of an external adverse pressure gradient flow over a flat plate. In the former, unsteadiness of the flow is captured and in the latter, the receptivity of the flow is studied in causing flow instability by triggering Tollmien-Schlichting waves. The new scheme shows a marked improvement over the explicit scheme for low Reynolds number steady flow in lid driven cavity. Whereas for the flow in the same geometry at higher Reynolds numbers, efficacy of the scheme is established by showing the formation of a triangular vortex and secondary vortical structures. Presented scheme is perfectly capable of expressing the diffusion operator accurately as shown via the capturing of instability waves inside the shear layer.

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