Nonlinear orbital stability of stationary discrete shock profiles for scalar conservation laws

TL;DR

The paper proves nonlinear orbital stability for spectrally stable stationary Lax discrete shock profiles in scalar conservation laws, showing that small perturbations in polynomially-weighted $\ell^1$ and $\ell^\infty$ spaces converge to a member of a one-parameter family of discrete shocks. The approach hinges on a precise fully discrete Green’s-function description of the linearization around the shock, with decay estimates that feed into a nonlinear induction argument via a quadratic remainder controlled by a mass-difference function. By combining spectral stability (via the Evans function) with diffusivity-induced decay, the authors obtain explicit rates in weighted norms and identify a mass parameter $\delta$ ensuring convergence to a discrete shock profile $\overline{u}^\delta$. The results extend Serre’s stability program to the fully discrete setting for scalar laws and outline steps toward analogous results for systems of conservation laws, while applying to a large family of dissipative numerical schemes that include artificial viscosity. Practically, this provides a robust framework for validating discrete-shock approximate solutions and their long-time behavior under small, mass-carrying perturbations in numerical simulations.

Abstract

For scalar conservation laws, we prove that spectrally stable stationary Lax discrete shock profiles are nonlinearly stable in some polynomially-weighted $\ell^1$ and $\ell^\infty$ spaces. In comparison with several previous nonlinear stability results on discrete shock profiles, we avoid the introduction of any weakness assumption on the amplitude of the shock and apply our analysis to a large family of schemes that introduce some artificial possibly high-order viscosity. The proof relies on a precise description of the Green's function of the linearization of the numerical scheme about spectrally stable discrete shock profiles obtained in [Coeu25]. The present article also pinpoints the ideas for a possible extension of this nonlinear orbital stability result for discrete shock profiles in the case of systems of conservation laws.

Figures (5)

• Figure 1: An example of stationary discrete shock profiles (SDSPs). The chosen scalar conservation law is Burger's equation $f(u):=\frac{u^2}{2}$. The numerical scheme is the modified Lax-Friedrichs scheme defined below as \ref{['def:MLF']} with $\boldsymbol{\nu}=0.5$ and $D=\frac{0.4}{\boldsymbol{\nu}}$ and the shock considered is the one associated with the states $u^-=1$ and $u^+=-1$. The red solution is a SDSP as well as all the blue ones. There exists a continuum of discrete shock profiles as stated in Hypothesis \ref{['H:SDSP']}. We also observe that Hypotheses \ref{['H:CVExpo']} and \ref{['H:identification']} seem to be verified.
• Figure 2: An example of nonlinear orbital stability of a discrete shock profile. On the left figure, we represent in blue a discrete shock profile $\overline{u}$ and in red a perturbation of the discrete shock profile $u^0:=\overline{u}+\textbf{h}$. The figures in the middle and on the right represent the evolution in time of the solution $u^n$ of the numerical scheme \ref{['def:SchemeNum']} with the initial condition $u^0$. We see that, in the long run, the sequence $(u^n)_{n\in\mathbb{N}}$ converges towards some other discrete shock profile $\overline{u}^\delta$ associated with the same shock. More precisely, the solution $u^n$ seems to converge towards the member of the family $(\overline{u}^\delta)_{\delta\in I}$ defined by the equality \ref{['IDMasseFinal']} since the sum of the differences between the elements of the red and blue sequences is conserved in time.
• Figure 3: In red, we have the curves $\mathcal{F}^\pm(\mathbb{S}^1)$. In gray, we represent the set $\mathcal{O}$ which corresponds to the unbounded component of $\mathbb{C}\backslash(\mathcal{F}^+(\mathbb{S}^1)\cup\mathcal{F}^-(\mathbb{S}^1))$. The elements of the set $\mathcal{O}$ are either eigenvalues of the operator ${\mathscr{L}}$ (represented in blue) or belong to the resolvent set $\rho({\mathscr{L}})$. We know that $1$ is an eigenvalue of ${\mathscr{L}}$. The first part of the spectral stability assumption described in Hypothesis \ref{['H:spec']} implies that the eigenvalues of ${\mathscr{L}}$ in $\mathcal{O}$ are located within the open unit disk.
• Figure 4: Representation of the values \ref{['Num:Sup']} for the choice of parameters \ref{['NumChoix1']} with $\textbf{p}=1$. The slopes that we obtain numerically are close to the expected slope, pointing to the fact that \ref{['Th:in']} seems sharp in this case.
• Figure 5: Representation of the values \ref{['Num:Sup']} for the choice of parameters \ref{['NumChoix2']} with $\textbf{p}=0.5$. For the figure on the right-side, the slope obtained is larger than one expected. However, if we performed the same calculations for $J_{max}$ and $n_{max}$ larger, the slope obtained would be closer to the expected one.

Theorems & Definitions (10)

• Lemma 1.1
• Lemma 1.2
• Proposition 1
• Theorem 1
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 4.2
• Lemma
• Lemma