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Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids

Wasilij Barsukow

TL;DR

The study analyzes how discontinuous Galerkin methods preserve discrete stationary states on Cartesian grids and how this relates to low Mach number behavior in the Euler equations. By translating the PDE stationary-state problem into a discrete evolution kernel via a Fourier-based framework, it shows that DG becomes stationarity preserving above a polynomial degree threshold, with the observed order of accuracy at stationary state depending sensitively on the numerical flux. Across fluxes, upwind/Rusanov often incur a one-order loss at steady state, central flux may fail for odd degrees, while a Low Mach flux can retain the optimal order, clarifying when low Mach fixes are necessary. The results provide a theoretical basis for the observed low Mach compliance of DG in the Euler limit and guide flux selection to preserve design accuracy for stationary states.

Abstract

Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of common finite volume methods for the Euler equations in the limit of low Mach number. In this work, the stationary states of the Discontinuous Galerkin (DG) method for linear acoustics on Cartesian grids are explored theoretically and experimentally, thus extending previous studies in the context of first-order finite difference methods. It is found that for a polynomial degree above some threshold, DG is stationarity preserving, but depending on the choice of numerical flux can suffer from a reduction of the order of accuracy at stationary state. This allows to explain the behaviour of the method for the Euler equations at low Mach number.

Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids

TL;DR

The study analyzes how discontinuous Galerkin methods preserve discrete stationary states on Cartesian grids and how this relates to low Mach number behavior in the Euler equations. By translating the PDE stationary-state problem into a discrete evolution kernel via a Fourier-based framework, it shows that DG becomes stationarity preserving above a polynomial degree threshold, with the observed order of accuracy at stationary state depending sensitively on the numerical flux. Across fluxes, upwind/Rusanov often incur a one-order loss at steady state, central flux may fail for odd degrees, while a Low Mach flux can retain the optimal order, clarifying when low Mach fixes are necessary. The results provide a theoretical basis for the observed low Mach compliance of DG in the Euler limit and guide flux selection to preserve design accuracy for stationary states.

Abstract

Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of common finite volume methods for the Euler equations in the limit of low Mach number. In this work, the stationary states of the Discontinuous Galerkin (DG) method for linear acoustics on Cartesian grids are explored theoretically and experimentally, thus extending previous studies in the context of first-order finite difference methods. It is found that for a polynomial degree above some threshold, DG is stationarity preserving, but depending on the choice of numerical flux can suffer from a reduction of the order of accuracy at stationary state. This allows to explain the behaviour of the method for the Euler equations at low Mach number.

Paper Structure

This paper contains 22 sections, 1 theorem, 116 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Low Mach number problems
  3. Relation to stationary states of linear acoustics
  4. Time integration and grid geometry
  5. Outline of the paper
  6. Behaviour of DG in the low Mach number regime
  7. The analysis of stationarity preservation on Cartesian grids
  8. The Fourier transform
  9. Cartesian grids and the discrete Fourier transform
  10. Discrete Fourier transform of the DG method
  11. Stationarity preservation of DG and the order of accuracy of stationary solutions
  12. General procedure
  13. Order of accuracy at steady state
  14. Upwind flux
  15. Rusanov flux
  16. ...and 7 more sections

Key Result

Theorem 4.1

Consider $q = \hat{Q} \exp(\mathbb{i} \mathbf k \cdot \mathbf x) \in \mathbb M^m_{\mathbf k}$. If $v = \widehat{\mathrm{DOF}}(\hat{Q}) + \mathcal{O}(\Delta x^{K+1})$, then $\| q - R v \|_{L^2} \in \mathcal{O}(\Delta x^{K+1})$.

Figures (16)

  • Figure 1: The low Mach number setup for the Euler equations, colour coded is the magnitude $|\mathbf v|$ of the velocity. Left: Grid of $25 \times 25$. Right: Grid of $50\times 50$.
  • Figure 2: Rusanov DG for the Euler equations with $\epsilon = 10^{-3}$, colour coded is the magnitude $|\mathbf v|$ of the velocity. Numerical solution at time $t=1$ at cell center is shown. Top: $K=0$, the solution is diffused away down to machine zero. Middle: $K=1$. Bottom: $K=2$. Left: Grid of $25 \times 25$. Right: Grid of $50\times 50$.
  • Figure 3:
  • Figure 4: Same as Figure \ref{['fig:eulerM3rusanov']}, but using the Roe flux.
  • Figure 5: Same as Figure \ref{['fig:eulerM3rusanov']}, but using the Low Mach flux \ref{['eq:euler0201']}. Top: $K=0$. Bottom: $K=1$.
  • ...and 11 more figures

Theorems & Definitions (11)

  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Theorem 4.1
  • proof
  • Definition 4.4
  • Example 4.1
  • Example 4.2
  • ...and 1 more