[02614] Convergence of a Second-Order Scheme for Nonlocal Traffic Flow Problems
Session Time & Room : 4E (Aug.24, 17:40-19:20) @E506
Type : Contributed Talk
Abstract : In this work, we focus on the construction and convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws to capture the interaction between drivers and the surrounding density of vehicles. Specifically, we combine MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme for these equations. We show that this scheme satisfies a maximum principle and obtain bounded variation estimates. Also, the scheme is shown to admit L1- Lipschitz continuity in time. Subsequently, employing the Kolmogorov's theorem with a modification and using a Lax-Wendroff type argument, the convergence of this scheme to the entropy solution of the underlying problem is established. Numerical examples are presented to validate our theoretical analysis. Additionally, we extend our analysis to two dimensional non-local problems, for which we present a positivity preserving second-order scheme. While first-order methods are typically reliable in computational fluid dynamics, higher-order methods can provide more accurate solutions at the same computational cost, especially for problems in two or three dimensions. Our proposed scheme thus has important implications for accurately approximating traffic flow equations, and our theoretical analysis provides a solid foundation for its practical implementation.
