Approximate deconvolution discretisation

Abstract

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding ‘filtered’ solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a ‘sharpening’ of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the ‘sharpening’ of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid.