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Nonlinear orbital stability of stationary shock profiles for the Lax-Wendroff scheme

Jean-François Coulombel, Grégory Faye

Abstract

In this article we study the spectral, linear and nonlinear stability of stationary shock profile solutions to the Lax-Wendroff scheme for hyperbolic conservation laws. We first clarify the spectral stability of such solutions depending on the convexity of the flux for the underlying conservation law. The main contribution of this article is a detailed study of the so-called Green's function for the linearized numerical scheme. As evidenced on numerical simulations, the Green's function exhibits a highly oscillating behavior ahead of the leading wave before this wave reaches the shock location. One of our main results gives a quantitative description of this behavior. Because of the existence of a one-parameter family of stationary shock profiles, the linearized numerical scheme admits the eigenvalue 1 that is embedded in its continuous spectrum, which gives rise to several contributions in the Green's function. Our detailed analysis of the Green's function describes these contributions by means of a so-called activation function. For large times, the activation function describes how the mass of the initial condition accumulates along the eigenvector associated with the eigenvalue 1 of the linearized numerical scheme. We can then obtain sharp decay estimates for the linearized numerical scheme in polynomially weighted spaces, which in turn yield a nonlinear orbital stability result for spectrally stable stationary shock profiles. This nonlinear result is obtained despite the lack of uniform ${\ell}$ 1 estimates for the Green's function of the linearized numerical scheme, the lack of such estimates being linked with the dispersive nature of the numerical scheme. This dispersive feature is in sharp contrast with previous results on the orbital stability of traveling waves or discrete shock profiles for parabolic perturbations of conservation laws.

Nonlinear orbital stability of stationary shock profiles for the Lax-Wendroff scheme

Abstract

In this article we study the spectral, linear and nonlinear stability of stationary shock profile solutions to the Lax-Wendroff scheme for hyperbolic conservation laws. We first clarify the spectral stability of such solutions depending on the convexity of the flux for the underlying conservation law. The main contribution of this article is a detailed study of the so-called Green's function for the linearized numerical scheme. As evidenced on numerical simulations, the Green's function exhibits a highly oscillating behavior ahead of the leading wave before this wave reaches the shock location. One of our main results gives a quantitative description of this behavior. Because of the existence of a one-parameter family of stationary shock profiles, the linearized numerical scheme admits the eigenvalue 1 that is embedded in its continuous spectrum, which gives rise to several contributions in the Green's function. Our detailed analysis of the Green's function describes these contributions by means of a so-called activation function. For large times, the activation function describes how the mass of the initial condition accumulates along the eigenvector associated with the eigenvalue 1 of the linearized numerical scheme. We can then obtain sharp decay estimates for the linearized numerical scheme in polynomially weighted spaces, which in turn yield a nonlinear orbital stability result for spectrally stable stationary shock profiles. This nonlinear result is obtained despite the lack of uniform 1 estimates for the Green's function of the linearized numerical scheme, the lack of such estimates being linked with the dispersive nature of the numerical scheme. This dispersive feature is in sharp contrast with previous results on the orbital stability of traveling waves or discrete shock profiles for parabolic perturbations of conservation laws.

Paper Structure

This paper contains 49 sections, 54 theorems, 711 equations, 16 figures.

Table of Contents

  1. Introduction
  2. The Lax-Wendroff scheme. Notation and main results
  3. The Lax-Wendroff scheme and its stationary shock profiles
  4. Main results
  5. Numerical experiments
  6. Spectral stability
  7. The Lopatinskii determinant
  8. The spatial Green's function
  9. Case I: $j_0 \ge 2$.
  10. Case II: $j_0=1$.
  11. Case III: $j_0=0$.
  12. Case IV: $j_0 \le -1$.
  13. Spectral stability. Proof of Theorem \ref{['thm1bis']}
  14. Instability cases
  15. Decomposing the spatial Green's function
  16. ...and 34 more sections

Key Result

Theorem 2.1

Let the shock shock satisfy the Rankine-Hugoniot condition RH and the entropy condition entropy. Let $\lambda$ satisfy the so-called CFL condition: Then there exist $\underline{\theta}>0$ and a one-parameter family of stationary discrete shock profiles $\mathbf{v}^\theta = (v_j^\theta)_{j \in {\mathbb Z}}$, $\theta \in (-\underline{\theta},\underline{\theta})$, that satisfies the following proper

Figures (16)

  • Figure 1: The characteristics on either side of the shock.
  • Figure 2: Discrete shock profiles for the Lax-Wendroff scheme applied to the Burgers equation. Left: the reference shock \ref{['shock']}. Middle: a discrete shock profile with same end states but negative mass difference ($\theta<0$ in Theorem \ref{['thm:Smyrlis']}). Right: a discrete shock profile with same end states but positive mass difference ($\theta>0$ in Theorem \ref{['thm:Smyrlis']}).
  • Figure 3: The evaluation at $j=0$ and $j=1$ of the family of stationary discrete shock profiles $\mathbf{v}^\theta$, $\theta \in {\mathbb R}$.
  • Figure 4: Evolution of a perturbation with zero mass of the reference discrete shock \ref{['shock']}. First line (from left to right): the initial condition, the solution at $n=40$, the solution at $n=150$. Second line (from left to right): the solution at $n=400$, the solution at $n=700$, the solution at $n=+\infty$ (convergence towards the reference shock \ref{['shock']}).
  • Figure 5: Evolution of a perturbation with zero mass of the reference discrete shock \ref{['shock']}.
  • ...and 11 more figures

Theorems & Definitions (86)

  • Theorem 2.1: Smyrlis Smyrlis
  • Theorem 2.2: Spectral stability for convex or concave fluxes
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['lem1']}
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.2
  • ...and 76 more