Multioperator high-order compact upwind methods for CFD parallel calculations

On Performance of Methods with Third- and Fifth-Order Compact Upwind Differencing

“…It turns out that their linear combinations can be used when constructing arbitrary-order schemes for parallel calculations [18]. Another interesting feature of CUD-II-5 is its ability to provide negligible phase errors practically for all wave lengths supported by grids when combined with the newly proposed CUD-based fifth-order two-step time integrators [19].…”

“…In the case of centered schemes one usually has |q| → 1 when Re c → ∞. It means that solution (18) can be completely erroneous due to ill-conditioned nature of difference equations. If, however, |q| is sufficiently large for Re c * ≤ Re ≤ ∞, one may hope that at least the "inviscid" part of the solution is reproduced with reasonable accuracy due to rapid decay of possible wiggles.…”

Development of arbitrary-order multioperators-based schemes for parallel calculations. 1: Higher-than-fifth-order approximations to convection terms

“…The idea of increasing approximation orders of numerical formulas by using appropriate linear combinations of operators from one-parameter operators families was first presented by the author at the Manchester parallel CFD conference (1997) [1]. It resulted from close examination of the Taylor expansion series for the earliest form of the compact upwind differencing (CUD) operators [2].…”

“…The potential for being basis operators satisfying (2) is an inherent feature of compact approximations having free parameters in their inverse operators. It is the case of the one-parametric families of compact upwind differencing (CUD) operators from [2] (their examples can be found also in [7]), the upwinding parameter s being viewed as the free parameter in (1). The third-order CUD operator was considered in the initial paper [1].…”

“…The concept of multioperators was introduced by the author in [1]. It was aimed at constructing prescribedorder approximations (formally, arbitrary-order ones) to various grid functionals by exploiting linear combinations of basis operators having relatively simple structures and depending on at least one free parameter.…”