Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws

Abstract

A new finite control volume in a Lagrangian-Eulerian framework is presented (see papers [1, 28]), in which a localspace-time domain is studied, in order to design a locally conservative scheme. Such scheme accounts for the delicate nonlinearbalance between the numerical approximations of the hyperbolic flux and the source term for balance law problems linked to thepurely hyperbolic character of conservation laws. Furthermore, by combining the ideas of this new approach, we give a formalconstruction of a new algorithm for solving several nonlinear hyperbolic conservation laws in two space dimensions. Here, a setof pertinent numerical experiments for distinct models is presented to evidence that we are calculating the correct qualitativelygood solutions.