An Analysis of the Riemann Problem for a $2 \times 2$ System of Keyfitz-Kranzer Type Balance Laws With a Time-Dependent Source Term

TL;DR

This paper analyzes the Riemann problem for a 2x2 Keyfitz–Kranzer–type balance/conservation system with a time-dependent source term, revealing non-self-similar, time-evolving wave structures that can traverse vacuum and a critical density $\rho=\overline{\rho}$. By converting to a conservative form using $\tilde{u}=u-\int_0^t a(s)ds$, the authors derive the eigenstructure, Rankine–Hugoniot conditions, and wave curves, identifying both classical waves and nonclassical delta shocks arising from degenerate hyperbolicity. They develop two approaches to delta shocks—the distributional RH framework and Nedeljkov’s shadow-wave method—and provide detailed conditions and ODEs governing delta-shock evolution, including special scaling regimes for $a<0$ and $a>0$. Numerical verification via the Local Lax–Friedrichs scheme illustrates moving admissibility regions, including multiple-region transitions and overcompressive regimes, and highlights how the time dependence yields region shifts over time. The work advances understanding of singular wave structures in degenerate, non-autonomous hyperbolic systems and lays groundwork for high-order, non-dissipative numerical methods for problems with vacuums and Delta solutions.

Abstract

We consider a system consisting of one conservation law and one balance law with a time-dependent source term, and provide a comprehensive analysis of Riemann solutions, including the non-classical overcompressive delta shocks. The minimal yet representative structure of the system captures essential features of transport under density constraints and, despite its simplicity, serves as a versatile prototype for crowd-limited transport processes across diverse contexts, including biological aggregation, ecological dispersal, granular compaction, and traffic congestion. In addition to non-self-similar solutions mentioned above, the associated Riemann problem admits solution structures that traverse vacuum states ($ρ= 0$) and the critical density threshold ($ρ= \barρ$), where mobility vanishes and characteristic speed degenerates. Moreover, the explicit time dependence in the source term leads to the breakdown of self-similarity, resulting in distinct Riemann solutions over successive time intervals and highlighting the dynamic nature of the solution landscape. The theoretical findings are numerically confirmed using the Local Lax-Friedrichs scheme.

Figures (24)

• Figure 1: Illustration of characteristics for an $a$-rarefaction curve.
• Figure 2: Case 18. $a = -0.5, a(t) = 0.1, \rho_L = 5, \overline{\rho} = 3, \tilde{u}_L = 4$.
• Figure 3: Case 13. $a = -0.5, a(t) = 0.1, \rho_L = 3, \overline{\rho} = 5, \tilde{u}_L = -4$.
• Figure 4: Case 19. $a = 0.5, a(t) = 0.1, \rho_L = 3, \overline{\rho} = 5, \tilde{u}_L = -4$.
• Figure 5: Case 1: State space of regions. $\lambda_a < 0 < \lambda_0$.

Theorems & Definitions (2)

• Remark 1
• Theorem 1