A unified framework to generate optimized compact finite difference schemes

Abstract

A unified framework to derive optimized compact schemes for a uniform grid is presented. The optimal scheme coefficients are determined analytically by solving an optimization problem to minimize the spectral error subject to equality constraints that ensure specified order of accuracy. A rigorous stability analysis for the optimized schemes is also presented. We analytically prove the relation between order of a derivative and symmetry or skew-symmetry of the optimal coefficients approximating it. We also show that other types of schemes e.g., spatially explicit, and biased finite differences, can be generated as special cases of the framework.

Spectrally-tuned compact finite-difference schemes with domain decomposition and applications to numerical relativity

“…The aim of this section is to provide a method to optimize finite-difference based numerical evaluation of an unknown derivative of a function based on known (derivative) data. Our approach builds upon the framework of [19] and extends it to allow incorporation of data from multiple derivatives in addition to function data in specification of stencils.…”

“…We can consider further introduction and constrained minimization against a well-defined objective function ε (e.g. characterizing some error metric of interest) so as to yield α (d K ) at some desired fixed formal order tuned against ε [19]. One can also impose constraints that fix the values of constants that appear in the truncation error ε T directly which is of use during domain-decomposition ( §II D and §II E).…”

“…This can be exploited through minimization against a suitably chosen cost function characterizing an error profile which allows for spectral-tuning [16][17][18]. A unified approach to this for construction of CFD schemes with arbitrary derivative degree, order, and stencil size (and bias) was recently introduced in [19]. The advantage of this latter approach in addition to the aforementioned generalizations is that various properties such as coefficient (skew-)symmetries do not need to be imposed a priori but are a consequence of the optimization procedure.…”

“…The advantage of this latter approach in addition to the aforementioned generalizations is that various properties such as coefficient (skew-)symmetries do not need to be imposed a priori but are a consequence of the optimization procedure. While quite general, the framework of [19] does not immediately embed derivation of multi-derivative such as those of [5,7] or Hermite-FD methods [20].…”

“…The first goal of this work is to obviate the need for repeated, hand construction of CFD schemes by automatizing the procedure of [19] numerically and extending it to multi-derivative schemes thereby allowing for rapid experimentation on model problems prior to general application. A second goal is to combine this framework with the BAA technique proposed in [8], extend it to centered schemes and consider how it may be further refined through an iterative procedure.…”