Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysis of a central MUSCL-type scheme for conservation laws with discontinuous flux

Nikhil Manoj, Sudarshan Kumar K

TL;DR

This work develops and analyzes a simple yet robust MUSCL-type central scheme, a Nessyahu-Tadmor–inspired central discretization, for scalar conservation laws with discontinuous flux $f(k(x),u)$ where $k$ has finite jump points. The authors establish convergence to a weak solution via compensated compactness, exploiting a priori cubic and quadratic estimates to obtain $W^{-1,2}_{loc}$-compactness despite the lack of BV bounds. To attain convergence to the entropy solution, they introduce a mesh-size dependent correction in the slope limiter within a predictor-corrector framework, enabling entropy convergence through a Vila-type argument adapted to discontinuous flux. Numerical experiments corroborate the theory, showing that the proposed SO scheme achieves lower diffusion and better resolution near flux discontinuities compared with a first-order LF baseline. The results provide a rigorous, Riemann-solver–free pathway to entropy-consistent second-order approximations for discontinuous-flux conservation laws, with clear avenues for extending to broader flux structures and higher dimensions.

Abstract

In this article, we propose a second-order central scheme of the Nessyahu-Tadmor-type for a class of scalar conservation laws with discontinuous flux and present its convergence analysis. Since solutions to problems with discontinuous flux generally do not belong to the space of bounded variation (BV), we employ the theory of compensated compactness to establish the convergence of approximate solutions. A major component of our analysis involves deriving the maximum principle and showing the $\mathrm{W}^{-1,2}_{\mathrm{loc}}$ compactness of a sequence constructed from approximate solutions. The latter is achieved through the derivation of several essential estimates on the approximate solutions. Furthermore, by incorporating a mesh-dependent correction term in the slope limiter, we show that the numerical solutions generated by the proposed second-order scheme converge to the entropy solution. Finally, we validate our theoretical results by presenting numerical examples.

Analysis of a central MUSCL-type scheme for conservation laws with discontinuous flux

TL;DR

This work develops and analyzes a simple yet robust MUSCL-type central scheme, a Nessyahu-Tadmor–inspired central discretization, for scalar conservation laws with discontinuous flux where has finite jump points. The authors establish convergence to a weak solution via compensated compactness, exploiting a priori cubic and quadratic estimates to obtain -compactness despite the lack of BV bounds. To attain convergence to the entropy solution, they introduce a mesh-size dependent correction in the slope limiter within a predictor-corrector framework, enabling entropy convergence through a Vila-type argument adapted to discontinuous flux. Numerical experiments corroborate the theory, showing that the proposed SO scheme achieves lower diffusion and better resolution near flux discontinuities compared with a first-order LF baseline. The results provide a rigorous, Riemann-solver–free pathway to entropy-consistent second-order approximations for discontinuous-flux conservation laws, with clear avenues for extending to broader flux structures and higher dimensions.

Abstract

In this article, we propose a second-order central scheme of the Nessyahu-Tadmor-type for a class of scalar conservation laws with discontinuous flux and present its convergence analysis. Since solutions to problems with discontinuous flux generally do not belong to the space of bounded variation (BV), we employ the theory of compensated compactness to establish the convergence of approximate solutions. A major component of our analysis involves deriving the maximum principle and showing the compactness of a sequence constructed from approximate solutions. The latter is achieved through the derivation of several essential estimates on the approximate solutions. Furthermore, by incorporating a mesh-dependent correction term in the slope limiter, we show that the numerical solutions generated by the proposed second-order scheme converge to the entropy solution. Finally, we validate our theoretical results by presenting numerical examples.
Paper Structure (18 sections, 18 theorems, 297 equations, 3 figures)

This paper contains 18 sections, 18 theorems, 297 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A brief review of conservation laws with discontinuous flux
  3. Scope and outline of the paper
  4. Preliminaries
  5. Notations
  6. Hypotheses
  7. Weak and entropy solutions
  8. Second-order central scheme
  9. Compensated compactness
  10. Maximum principle and L-infinity-stability
  11. A priori estimates
  12. Convergence of the second-order scheme
  13. Convergence to the entropy solution
  14. Numerical experiments
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 4.1

\newlabellemma:compcompactness0 Assume that the hypotheses hyp:H1-hyp:H5 hold true. Let $\{u^\varepsilon\}_{\varepsilon > 0}$ be a sequence of measurable functions defined on $\mathbb{R} \times \mathbb{R}^+$ that satisfies the following two conditions: Then, there exists a subsequence of $\{u^\varepsilon\}_{\varepsilon > 0}$ that converges pointwise a.e. to a function $u \in \mathrm{L}^{\infty}(

Figures (3)

  • Figure 1: The fluxes to the left ($f_l$) and right ($f_r$) of $x=0.$
  • Figure 2: Example \ref{['example:multipl_flux']}. Numerical solutions obtained by evolving \ref{['eq:karlsen_ic']} with $\Delta x = 0.04$ and $\Delta t = 1/750.$
  • Figure 3: Example \ref{['example:mishra2017']}. Numerical solutions obtained by evolving \ref{['eq:icdflu']} with $\Delta x = 0.16$ and $\Delta t = 0.008.$

Theorems & Definitions (42)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 3.1
  • Theorem 4.1: Compensated compactness theorem
  • Lemma 4.2
  • Theorem 5.1
  • Proof 1
  • Lemma 6.1
  • ...and 32 more