The following work have been accepted for publication on "Numerical Methods for Partial Differential Equations" and it will available online soon (MNPDE, https://onlinelibrary.wiley.com/journal/10982426):

An enhanced Lagrangian-Eulerian method for a class of balance Laws: numerical analysis via a weak asymptotic method with applications

Eduardo Abreu, Eduardo Pandini and Wanderson José Lambert.

Highlights:
In this work, we designed and implemented an enhanced Lagrangian-Eulerian numerical method for solving a wide range of nonlinear balance laws, including systems of hyperbolic equations with source terms. We developed both fully discrete and semi-discrete formulations, and extended the concept of No-Flow curves to this general class of nonlinear balance laws. We conducted a numerical convergence study using weak asymptotic analysis, which involved investigating the existence, uniqueness, and regularity of entropy-weak solutions computed with our scheme. The proposed method is Riemann-solver-free. To evaluate the shock capturing capabilities of the enhanced Lagrangian-Eulerian  numerical scheme, we carried out numerical experiments that demonstrate its ability to accurately resolve the key features of balance law models and hyperbolic problems. A representative set of numerical examples is provided to illustrate the accuracy and robustness of the proposed method.