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Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

Robert Altmann, Jan Heiland

TL;DR

This work treats the incompressible Navier–Stokes equations from a differential-algebraic equations (DAE) perspective, emphasizing how spatial discretization induces a coupled velocity–pressure constraint that yields strangeness in the discrete setting. It shows that common time-stepping schemes (e.g., projection, SIMPLE, and artificial compressibility) produce index-1 difference-algebraic equations, effectively removing strangeness, while naive implicit–explicit Euler can yield index-2 behavior and potential instability under inexact solves. The authors develop and compare index-reduction strategies (penalty, derivative of constraint, minimal extension) and illustrate their impact on stability and accuracy via a 2D cylinder wake benchmark, providing reproducible code. The results offer guidance for robust NSE time integration and contribute a rigorous index-theoretic lens for evaluating discretization choices in computational fluid dynamics.

Abstract

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.

Continuous, Semi-discrete, and Fully Discretized Navier-Stokes Equations

TL;DR

This work treats the incompressible Navier–Stokes equations from a differential-algebraic equations (DAE) perspective, emphasizing how spatial discretization induces a coupled velocity–pressure constraint that yields strangeness in the discrete setting. It shows that common time-stepping schemes (e.g., projection, SIMPLE, and artificial compressibility) produce index-1 difference-algebraic equations, effectively removing strangeness, while naive implicit–explicit Euler can yield index-2 behavior and potential instability under inexact solves. The authors develop and compare index-reduction strategies (penalty, derivative of constraint, minimal extension) and illustrate their impact on stability and accuracy via a 2D cylinder wake benchmark, providing reproducible code. The results offer guidance for robust NSE time integration and contribute a rigorous index-theoretic lens for evaluating discretization choices in computational fluid dynamics.

Abstract

The Navier--Stokes equations are commonly used to model and to simulate flow phenomena. We introduce the basic equations and discuss the standard methods for the spatial and temporal discretization. We analyse the semi-discrete equations -- a semi-explicit nonlinear DAE -- in terms of the strangeness index and quantify the numerical difficulties in the fully discrete schemes, that are induced by the strangeness of the system. By analyzing the Kronecker index of the difference-algebraic equations, that represent commonly and successfully used time stepping schemes for the Navier--Stokes equations, we show that those time-integration schemes factually remove the strangeness. The theoretical considerations are backed and illustrated by numerical examples.

Paper Structure

This paper contains 30 sections, 1 theorem, 90 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continuous model
  3. Derivation of the Navier--Stokes equations
  4. Formulation as operator DAE
  5. Semi-discrete equations
  6. Spatial discretization by finite elements
  7. Finite volumes and finite differences
  8. Index of the DAE
  9. Fully discrete approximation schemes
  10. Index and causality of discrete systems
  11. Inherent instabilities of $\Delta$AE s of higher index
  12. Common time-stepping schemes as index-1 $\Delta$AEs
  13. Projection
  14. SIMPLE scheme -- implicit pressure correction
  15. Artificial compressibility
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $W\colon I \to \mathbb R^{3}$ describe a smoothly moving control volume and let $f \colon t \times W(t) \mapsto \mathbb R^{}$ be a sufficiently smooth function, then

Figures (5)

  • Figure 5.1: Illustration of the geometrical setup including the domains of distributed control and observation and of the velocity magnitude for the cylinder wake. Figure taken from BehBH17.
  • Figure 5.2: Snapshots of the velocity magnitude computed with SIMPLE with $\tau = 1/1024$ and exact solves taken at $t=0$ (top) and $t=1$ (bottom).
  • Figure 5.3: Link to code and data.
  • Figure 5.4: Error in the velocity and the pressure approximation provided by the implicit-explicit Euler algorithm for iterative solves with varying tolerances. The crosses are the errors obtained with direct solves.
  • Figure 5.5: Error in the velocity and the pressure approximation provided by the SIMPLE algorithm for exact solves and iterative solves with varying tolerances tol.

Theorems & Definitions (15)

  • Theorem 2.1: see Rey03, Eq.(16) and ChoM93, p.10
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 4.1
  • Definition 4.2
  • Definition 4.3: Dai89
  • ...and 5 more