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Convergence of a Finite Volume Scheme for Compactly Heterogeneous Scalar Conservation Laws

Abraham Sylla

TL;DR

The paper develops a finite volume scheme for a scalar conservation law with a space-dependent, compactly supported flux and proves convergence to the Kruzhkov entropy solution without relying on Kruzhkov’s growth condition. It leverages a discontinuous-flux framework to discretize space and solve edge-wise Riemann problems, constructing discrete steady states to obtain uniform L∞ bounds. Convergence is established via compensated compactness, with careful handling of entropy productions and discrete entropy inequalities. The work provides an alternative existence route for the Cauchy problem under compact heterogeneity and broad flux classes, with potential impact on modeling problems featuring spatially varying flux functions.

Abstract

We build a finite volume scheme for the scalar conservation law $\partial_t u + \partial_x (H(x, u)) = 0$ with bounded initial condition for a wide class of flux function $H$, convex with respect to the second variable. The main idea for the construction of the scheme is to use the theory of discontinuous flux. We prove that the resulting approximating sequence converges boundedly almost everywhere on $\mathopen]0, +\infty\mathclose[$ to the entropy solution.

Convergence of a Finite Volume Scheme for Compactly Heterogeneous Scalar Conservation Laws

TL;DR

The paper develops a finite volume scheme for a scalar conservation law with a space-dependent, compactly supported flux and proves convergence to the Kruzhkov entropy solution without relying on Kruzhkov’s growth condition. It leverages a discontinuous-flux framework to discretize space and solve edge-wise Riemann problems, constructing discrete steady states to obtain uniform L∞ bounds. Convergence is established via compensated compactness, with careful handling of entropy productions and discrete entropy inequalities. The work provides an alternative existence route for the Cauchy problem under compact heterogeneity and broad flux classes, with potential impact on modeling problems featuring spatially varying flux functions.

Abstract

We build a finite volume scheme for the scalar conservation law with bounded initial condition for a wide class of flux function , convex with respect to the second variable. The main idea for the construction of the scheme is to use the theory of discontinuous flux. We prove that the resulting approximating sequence converges boundedly almost everywhere on to the entropy solution.
Paper Structure (18 sections, 18 theorems, 117 equations)

This paper contains 18 sections, 18 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Description of the scheme and main result
  3. Proofs
  4. Discontinuous Flux
  5. If $(u_l, u_r) \in \mathcal{G}$,
  6. Case (I)
  7. Case (II)
  8. Case (III)
  9. Case (IV)
  10. Steady states and stability
  11. Claim 1:
  12. Claim 2:
  13. Convergence of the scheme
  14. Compensated compactness
  15. Convergence
  16. ...and 3 more sections

Key Result

Theorem 1.2

CPS2023 Assume that $H$ satisfies eq:smoothness--eq:LocalizedSpace--eq:convexity. Then for all $u_o \in \mathbf{L}^{\infty}(\mathbb{R}, \mathbb{R})$, the Cauchy problem eq:CL admits a unique entropy solution $u \in \mathbf{L}^{\infty}(\mathopen] 0, +\infty \mathclose[\times \mathbb{R},\mathbb{R})$.

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 2.1
  • Remark 2.1
  • Remark 3.1
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.2
  • ...and 16 more