Linear stability of discrete shock profiles for systems of conservation laws

TL;DR

The paper proves linear orbital stability of spectrally stable stationary discrete shock profiles for conservative finite-difference schemes solving systems of conservation laws, under a sharp spectral assumption rather than small-amplitude restrictions. Central to the argument is a refined, pointwise description of the temporal and spatial Green’s functions, obtained by extending Lafitte-Godillon’s geometric-dichotomy framework and Evans-function analysis to a broad class of odd-order schemes with numerical viscosity. The authors decompose the Green’s function into competing traveling Gaussian waves, reflections, transmissions, and a non-decaying center mode, then derive sharp decay estimates for the associated semigroup on weighted l^r spaces, establishing linear stability (Theorem thStab). The results offer a path toward nonlinear stability for discrete shock profiles and provide a versatile toolkit for analyzing fully discrete schemes beyond the modified Lax-Friedrichs case, including higher-order schemes with viscosity. The combination of spectral and Green’s-function techniques offers precise, uniform control of the discrete dynamics and highlights the role of the Evans function and geometric-dichotomy machinery in a fully discrete setting.

Abstract

We prove the linear orbital stability of spectrally stable stationary discrete shock profiles for conservative finite difference schemes applied to systems of conservation laws. The proof relies on an accurate description of the pointwise asymptotic behavior of the Green's function associated with those discrete shock profiles, improving on the result of Lafitte-Godillon [God03]. The main novelty of this stability result is that it applies to a fairly large family of schemes that introduce some artificial possibly high-order viscosity. The result is obtained under a sharp spectral assumption rather than by imposing a smallness assumption on the shock amplitude.

Figures (6)

• Figure 1: In red, we have the spectrum of the operators $\mathscr{L}^\pm$ which corresponds to the union of the curves $\mathcal{F}_l^\pm(\mathbb{S}^1)$. Here we chose $d=1$, i.e. one curve corresponds to the spectrum of $\mathscr{L}^+$ and the other corresponds to the spectrum of $\mathscr{L}^-$. In gray, we represent the set $\mathcal{O}$ which corresponds to the unbounded component of $\mathbb{C}\backslash(\sigma(\mathscr{L}^+)\cup\sigma(\mathscr{L}^-))$. The elements of the set $\mathcal{O}$ are either eigenvalues of the operator $\mathscr{L}$ (represented by green dots) or belong to the resolvent set $\rho(\mathscr{L})$. We know that $1$ is an eigenvalue of $\mathscr{L}$ and Hypothesis \ref{['H:spec']} implies that the eigenvalues of $\mathscr{L}$ in $\mathcal{O}$ are located within the open unit disk.
• Figure 2: A schematic representation of the result of Theorem \ref{['th:Green']} on the Green's function $\mathscr{G}(n,j_0,j)$. Here we represent a case where $j_0\in\mathbb{N}$, $d=4$ and $I=3$. We recall that the integer $I$ is defined by Hypothesis \ref{['H:Lax']}. We have $d$ generalized Gaussian waves (in blue) arising from the Dirac mass at $j_0$ which travel along the characteristics of the right state $u^+$. The ones reaching the shock location are decomposed into reflected waves (in purple), transmitted waves (in green) and the activation of the component of the Green's function along the vector space $\ker(Id_{\ell^2}-\mathscr{L})$ (in red). We only represent this decomposition for one of the incoming waves.
• Figure 3: A representation of the path described in \ref{['def:Paths']}: $\Gamma_{out}(\eta)$, $\Gamma_{in}^\pm(\eta)$, $\Gamma_{in}^0(\eta)$ and $\Gamma_{d}(\eta)$
• Figure 4: Solution $u^n$ of the numerical scheme \ref{['num:MLF']} for the initial condition \ref{['condini']}. The solution $u^n$ seems to converge in time (quite fast) towards a stationary discrete shock profile $\overline{u}^s$.
• Figure 5: Representation of the Green's function $\mathscr{G}(n,j_0,j)$ for $j_0=50$. The initial time $n=0$ has been removed for readability since the Green's function is equal to the sequence $\delta_{j_0}$ at this time step.

Theorems & Definitions (42)

• Proposition 1
• Lemma 1.1
• Lemma 1.2
• Theorem 1
• Theorem 2
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3: Spectral Splitting
• Lemma 3.4: Geometric dichotomy
• Lemma 3.5