A Godunov--type scheme for a scalar conservation law with space-time flux discontinuity

TL;DR

This work tackles a scalar conservation law with a space-time flux discontinuity along a moving turning curve $\xi(t)$, formulated as $F(t,x,\rho)=\mathrm{sign}(x-\xi(t)) f(\rho)$ with a concave $f$. A Godunov-type finite-volume scheme is developed that employs a moving mesh around $\xi$ and a nonclassical Riemann solver at the interface to respect Rankine–Hugoniot conditions, while standard fluxes apply away from the interface. The authors establish a rigorous convergence framework based on the germ of admissible solutions, proving uniform $L^\infty$ bounds and discrete entropy inequalities that lead to convergence to a $\widetilde{\mathcal{G}}_{\alpha}$-entropy process solution under suitable CFL conditions. Numerical experiments with $f(\rho)=\rho(1-\rho)$ and a moving turning curve demonstrate accurate capture of interface interactions, suppression of spurious oscillations, and sublinear convergence rates around 0.8, validating the method’s robustness and accuracy in complex flux-interface scenarios.

Abstract

We present and analyze a new finite volume scheme of Gudonov-type for a nonlinear scalar conservation law whose flux function has a discontinuous coefficient due to time-dependent changes in its sign along a Lipschitz continuous curve.

Figures (7)

• Figure 2: Illustration of mesh adaptation for Case A
• Figure 3: Case B. Dashed lines indicate that the corresponding cell boundary is removed.
• Figure 4: Approximate solution of Examples A (left) and B (right) in the $x-t$ plane.
• Figure 5: Evolution of the approximate solution of Examples A in the $\rho-x$ plane for $t=0,~0.45, ~0.55,$ and $1.0$.
• Figure 6: Evolution of the approximate solution of Examples C in the $\rho-x$ plane for $t=0,~0.095, ~0.45,$ and $0.92$.

Theorems & Definitions (29)

• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.1
• Remark 1
• Definition 2.3
• Definition 2.4
• Proposition 2.2
• proof
• Lemma 2.5