Convergence of a Lagrangian-Eulerian scheme by a weak asymptotic analysis for one-dimensional hyperbolic problems

TL;DR

This paper introduces a new fully discrete Lagrangian--Eulerian scheme for one-dimensional hyperbolic problems, demonstrating its convergence, bounded variation properties, and effectiveness in capturing nonlinear wave interactions through weak asymptotic analysis.

Contribution

It develops a novel Lagrangian--Eulerian scheme with weak asymptotic analysis, providing theoretical convergence results and practical algorithms for hyperbolic PDEs.

Findings

Proven convergence of the scheme to entropy solutions.

Effective in modeling shock interactions and rarefaction waves.

Validated using Wasserstein distance in numerical experiments.

Abstract

In this paper, we study both convergence and bounded variation properties of a new fully discrete conservative Lagrangian--Eulerian scheme to the entropy solution in the sense of Kruzhkov (scalar case) by using a weak asymptotic analysis. We discuss theoretical developments on the conception of no-flow curves for hyperbolic problems within scientific computing. The resulting algorithms have been proven to be effective to study nonlinear wave formations and rarefaction interactions. We present experiments to a study based on the use of the Wasserstein distance to show the effectiveness of the no-flow curves approach in the cases of shock interaction with an entropy wave related to the inviscid Burgers' model problem and to a 2x2 nonlocal traffic flow symmetric system of type Keyfitz--Kranzer.