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Solving Random Hyperbolic Conservation Laws Using Linear Programming

Shaoshuai Chu, Michael Herty, Maria Lukacova-Medvidova, Yizhou Zhou

TL;DR

This paper addresses the challenge of solving random nonlinear hyperbolic conservation laws by formulating a measure-valued problem using Young measures, which yields a linear PDE in the measure space. A linear-programming closure is constructed to determine a family of measures that minimize entropy while matching a finite set of stochastic moments, providing a flux closure that preserves key structures of the underlying system. The authors develop semi-discrete and fully discrete finite-volume schemes around this LP-based closure and demonstrate results on the Burgers equation and the isentropic Euler system, including a test with discontinuous flux that shows entropy-driven selection of weak solutions. The method offers a unifying, more structure-preserving alternative to stochastic Galerkin and Monte Carlo approaches, with clear trade-offs in intrusiveness and computational cost, and it highlights the role of entropy in selecting physically relevant measure-valued solutions.

Abstract

A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial differential equation with respect to the Young measure and allows to compute the approximation based on linear programming problems. We analyze structure-preserving properties of the derived numerical method and discuss its advantages and disadvantages. We numerically demonstrate the approach on the one-dimensional Burgers and isentropic Euler equations and compare with stochastic collocation. In addition, we introduce a discontinuous-flux test in which different entropies used in the linear-program objective select different weak entropy solutions, and we report the corresponding changes in the moments and supports of the Young measure.

Solving Random Hyperbolic Conservation Laws Using Linear Programming

TL;DR

This paper addresses the challenge of solving random nonlinear hyperbolic conservation laws by formulating a measure-valued problem using Young measures, which yields a linear PDE in the measure space. A linear-programming closure is constructed to determine a family of measures that minimize entropy while matching a finite set of stochastic moments, providing a flux closure that preserves key structures of the underlying system. The authors develop semi-discrete and fully discrete finite-volume schemes around this LP-based closure and demonstrate results on the Burgers equation and the isentropic Euler system, including a test with discontinuous flux that shows entropy-driven selection of weak solutions. The method offers a unifying, more structure-preserving alternative to stochastic Galerkin and Monte Carlo approaches, with clear trade-offs in intrusiveness and computational cost, and it highlights the role of entropy in selecting physically relevant measure-valued solutions.

Abstract

A novel structure-preserving numerical method to solve random hyperbolic systems of conservation laws is presented. The method uses a concept of generalized, measure-valued solutions to random conservation laws. This yields a linear partial differential equation with respect to the Young measure and allows to compute the approximation based on linear programming problems. We analyze structure-preserving properties of the derived numerical method and discuss its advantages and disadvantages. We numerically demonstrate the approach on the one-dimensional Burgers and isentropic Euler equations and compare with stochastic collocation. In addition, we introduce a discontinuous-flux test in which different entropies used in the linear-program objective select different weak entropy solutions, and we report the corresponding changes in the moments and supports of the Young measure.
Paper Structure (15 sections, 3 theorems, 52 equations, 8 figures, 2 tables)

This paper contains 15 sections, 3 theorems, 52 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Method
  3. Minimization Formulation for Stochastic Galerkin Approach
  4. Formal Derivation of the Proposed Scheme
  5. Semi-Discrete Scheme with the Lax-Friedrichs Flux
  6. Summary and Discussion of Proposed Method
  7. Fully Discrete Scheme
  8. Properties of the Fully Discrete Scheme
  9. Numerical Results
  10. Convergence Studies for the Burgers Equation
  11. Burgers Equation with Sinusoidal Initial Data
  12. Burgers Equation with Non-Atomic Support
  13. Riemann Problem for the Isentropic Euler System
  14. Conservation Laws with Discontinuous Flux
  15. Discussion and Outlook

Key Result

Lemma 2.1

Provided that the feasible set given by linprog-a--linprog-e is non--empty, the linear program linprog has a solution.

Figures (8)

  • Figure 1: Numerical solutions obtained by the stochastic collocation Lax-Friedrichs method (left) and the proposed method (right). Initial data are $u(0,x,\xi)=\xi \sin(2 \pi x).$ The final time $T=1/4.$
  • Figure 2: Numerical solution obtained by the proposed method with modified cost functional in the linear program \ref{['linprog']}. Initial data are $u(0,x,\xi)=\xi \sin(2 \pi x).$ The final time is $T=1/4.$
  • Figure 3: Young measure at the initial time $t = 0$ (left) and its logarithmic values (right).
  • Figure 4: Young measure at the final time $T = 0.25$ (left) and its logarithmic values (right) with $\lambda_F=0.05$.
  • Figure 5: Young measure at the final time $T = 0.25$ (left) and its logarithmic values (right) with $\lambda_F=1$.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 1 more