An Enhanced Lagrangian‐Eulerian Method for a Class of Balance Laws: Numerical Analysis via a Weak Asymptotic Method With Applications

ABSTRACT 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.