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

Abstract

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.