Improving spatial resolution characteristics of finite difference and finite volume schemes by approximate deconvolution pre-processing

A pre-processing step is proposed as a general method to enhance resolution properties of low order numerical differentiation and interpolation. Pre-processing operators are designed by taking two or more terms in the approximate deconvolution formula and using a local filter whose response characteristics are close to those of the numerical operation considered; operators for second order central differencing for first and second derivatives and also for a finite volume method are determined. In addition to the higher resolution the effective order of the truncation error can also be increased. The repeated filtering operations couple the operation over a wider stencil without using direct formulas. The effect of improving the resolution properties is illustrated first by computing the propagation of a square wave by a 1D, linear convection equation. Next, pre-processing is implemented in a standard finite volume code for solving Navier-Stokes equations for incompressible flow. In the Taylor-Green vortex flow, the improvements in order behaviour are demonstrated. The instability of plane Poiseuille flow, which is a sensitive test of resolution ability, shows that the predicted growth rates with the pre-processing scheme are more accurate than those obtained with a second order scheme. Direct numerical simulations of turbulent channel flow show that the improvement allows accurate solutions to be found on smaller grids.