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Numerical Analysis of Stabilization for Random Hyperbolic Systems of Conservation Laws

Shaoshuai Chu, Michael Herty, Alexander Kurganov

TL;DR

This work extends deterministic Lyapunov-based stabilization to random 1-D hyperbolic systems of conservation laws by introducing a stochastic discrete Lyapunov function within a finite-volume discretization. It derives explicit exponential decay rates for linear systems, dependent on boundary-control parameters, grid resolution, and statistics of random inputs, and validates these rates numerically for linear advection, linearized Saint-Venant, and nonlinear perturbations. Extensions to broader system classes and boundary conditions are presented, with rigorous decay proofs for the linear and certain balanced cases. Numerical experiments compare first-order upwind and second-order central-upwind schemes, showing that higher-order discretizations yield sharper decay estimates while preserving stability. The results establish a practical framework for stabilizing random hyperbolic balance laws and point toward future work on nonlinear and nonconservative systems and high-order methods.

Abstract

This paper extends the deterministic Lyapunov-based stabilization framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite volume discretization method from [{\sc M. Banda and M. Herty}, Math. Control Relat. Fields., 3 (2013), pp. 121--142], we introduce a stochastic discrete Lyapunov function to prove the exponential decay of numerical solutions for systems with random perturbations. For linear systems, we derive explicit decay rates, which depend on boundary control parameters, grid resolutions, and the statistical properties of the random inputs. Theoretical decay rates are verified through numerical examples, including boundary stabilization of the linear wave equations and linearized shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearized Saint-Venant system with source terms.

Numerical Analysis of Stabilization for Random Hyperbolic Systems of Conservation Laws

TL;DR

This work extends deterministic Lyapunov-based stabilization to random 1-D hyperbolic systems of conservation laws by introducing a stochastic discrete Lyapunov function within a finite-volume discretization. It derives explicit exponential decay rates for linear systems, dependent on boundary-control parameters, grid resolution, and statistics of random inputs, and validates these rates numerically for linear advection, linearized Saint-Venant, and nonlinear perturbations. Extensions to broader system classes and boundary conditions are presented, with rigorous decay proofs for the linear and certain balanced cases. Numerical experiments compare first-order upwind and second-order central-upwind schemes, showing that higher-order discretizations yield sharper decay estimates while preserving stability. The results establish a practical framework for stabilizing random hyperbolic balance laws and point toward future work on nonlinear and nonconservative systems and high-order methods.

Abstract

This paper extends the deterministic Lyapunov-based stabilization framework to random hyperbolic systems of conservation laws, where uncertainties arise in boundary controls and initial data. Building on the finite volume discretization method from [{\sc M. Banda and M. Herty}, Math. Control Relat. Fields., 3 (2013), pp. 121--142], we introduce a stochastic discrete Lyapunov function to prove the exponential decay of numerical solutions for systems with random perturbations. For linear systems, we derive explicit decay rates, which depend on boundary control parameters, grid resolutions, and the statistical properties of the random inputs. Theoretical decay rates are verified through numerical examples, including boundary stabilization of the linear wave equations and linearized shallow-water flows with random perturbations. We also present the decay rates for a nonlinear example and for the linearized Saint-Venant system with source terms.
Paper Structure (28 sections, 7 theorems, 107 equations, 8 tables)

This paper contains 28 sections, 7 theorems, 107 equations, 8 tables.

Table of Contents

  1. Introduction
  2. An Overview of the Studied Hyperbolic Systems
  3. The Discretization Scheme
  4. Extensions of Theorem \ref{['thm1']}
  5. Extension of Theorem \ref{['thm1']} to the Case of Linear $F$
  6. Extension of Theorem \ref{['thm1']} to the Case $m \ne p$
  7. Extension of Theorem \ref{['thm1']} to Different Boundary Conditions
  8. Numerical Results
  9. Boundary Stabilization of the Linear Advection Equation
  10. Example 1.
  11. Example 2.
  12. Boundary Stabilization of the Linearized Saint-Venant System
  13. Example 3.
  14. Example 4.
  15. Example 5.
  16. ...and 13 more sections

Key Result

Proposition 3.1

Assume that $F$ is diagonal for all $\bm{u}\in B_\varepsilon(0) \subset \mathbb{R}^p$ where $B_\varepsilon(0)$ is an open ball in $\mathbb{R}^p$ centered at the origin and with radius $\varepsilon$. Also assume that and the boundary conditions are prescribed as where If $\max\limits_{i=1,\ldots,m} \kappa_i <1$, then the equilibrium $\bm{u} \equiv 0$ for (1.2) with (1.5) and (1.6) is exponentia

Theorems & Definitions (12)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Remark 4.1
  • ...and 2 more