On Consistency of Finite Difference Approximations to the Navier-Stokes Equations

Abstract

Abstract. In the given paper, we confront three finite difference approximations to the Navier-Stokes equations for the two-dimensional viscous incomressible fluid flows. Two of these approximations were generated by the computer algebra assisted method proposed based on the finite volume method, numerical integration, and difference elimination. The third approximation was derived by the standard replacement of the temporal derivatives with the forward differences and the spatial derivatives with the central differences. We prove that only one of these approximations is strongly consistent with the Navier-Stokes equations and present our numerical tests which show that this approximation has a better behavior than the other two.

Algorithmic Approach to Strong Consistency Analysis of Finite Difference Approximations to PDE Systems

“…(16) and (18) (19) and (21) give s-consistent FDA of the Navier-Stokes and pressure Poisson equations in the two-dimensional case as well. Examples of such FDA were studied in [6]. One more s-consistent two-dimensional FDA was derived in [7].…”

“…In [4,5], for polynomially nonlinear PDE systems and regular solution grids, we introduced the novel concept of strong consistency, or s-consistency, which strengthens the concept of consistency and means that any element of the perfect difference ideal generated by the polynomials in FDA approximates an element in the radical differential ideal generated by the polynomials in PDE. In the subsequent papers [6,7], by computational experiments with two-dimensional incompressible Navier-Stokes equations, it was shown that s-consistent FDA have much better numerical behavior than FDA which are not s-consistent.…”

Nonlinear Waves Mathematical Modeling in Coaxial Shells Filled with Viscous Liquid

“…В работах [11,12] развит подход к построению разностных схем, основанный на построении пере-определенной системы разностных уравнений, получаемой из аппроксимации интегральных законов сохранения и интегральных соотношений, связывающих искомые функции и их производные. По-строение базиса Грёбнера соответствующих разностных многочленов позволяет построить разност-ную схему, которая автоматически обеспечивает выполнение интегральных законов сохранения по областям, составленным из шаблонов интегрирования.…”

A Strongly Consistent Finite Difference Scheme for Steady Stokes Flow and its Modified Equations

“…The last criterion was designed in [12] for linear PDE systems and then generalized in [9] to polynomially nonlinear systems. The computational experiments done in papers [2,3] with the Navier-Stokes equations demonstrated a substantial superiority in numerical behavior of s-consistent schemes over s-inconsistent ones.…”

“…In this paper, we consider the two-dimensional flow of an incompressible fluid described by the following system of partial differential equations (PDEs): Here the velocities u and v, the pressure p, and the external forces f (1) and f (2) are functions in x and y; Re is the Reynolds number and ∆ := ∂ xx + ∂ yy is the Laplace operator. A flow that is governed by these equations is denoted in the literature as a Stokes flow or a creeping flow.…”