Systems of Nonlocal Conservation Laws with Memory and Their Zero Retention Limit

TL;DR

This paper investigates nonlocal conservation laws with memory effects, establishing existence, uniqueness, and asymptotic behavior of entropy solutions, along with convergence rates for numerical schemes as memory effects diminish.

Contribution

It provides the first convergence and asymptotic analysis of finite volume schemes for nonlocal conservation laws with memory, without geometric restrictions.

Findings

Proved existence and uniqueness of entropy solutions.

Demonstrated convergence to memoryless nonlocal conservation laws.

Derived convergence rate estimates for numerical schemes.

Abstract

We study the entropy solution for a class of systems of nonlocal conservation laws in which the convective flux is convoluted with a kernel in both spatial and temporal variables. This formulation models the flux dependence on the solution within its spatial neighbourhood (nonlocal in space) as well as on prior states in time (nonlocal in time), thereby incorporating memory effects. In addition, employing a convergent finite volume approximation, the existence of the entropy solution is discussed. The uniqueness of such entropy solutions is also established. In addition, we analyze the asymptotic behaviour of the solutions as the support of the temporal convolution kernel shrinks, demonstrating the "memory-to-memoryless" effect and convergence to the entropy solution of the corresponding nonlocal conservation law without memory (i.e., nonlocal only in space). Convergence rate estimates are derived. In addition, the proposed numerical approximations are shown to be asymptotically compatible with this passage to the memoryless limit by deriving the corresponding asymptotic convergence rate estimates. The analysis is carried out in a very general setting, without imposing any geometric restrictions such as the convexity of the spatial and temporal convolution kernels, unlike the existing literature on the asymptotic analysis of nonlocal-in-space only conservation laws. To the best of our knowledge, this provides the first convergence and asymptotic analysis for finite volume schemes applied to nonlocal conservation laws with memory. Numerical experiments are included to illustrate the theory.