Solving hyperbolic conservation laws by using Lagrangian-Eulerian approach

We discuss a procedure for numerically solving nonlinear hyperbolic conserva- tion law problems by means of a Lagrangian-Eulerian framework. The underlying hyperbolic conservation law is written in a space-time divergence form, so that inherent conservation properties of the problem are reflected in the numerical scheme. In order to enhance resolution and accuracy of the approximations, we make use of polynomial reconstruction ideas into the Lagrangian-Eulerian novel approach. Finally, numerical results are given to verify the formal construction as well as to demonstrate its accuracy, efficiency, and versatility. These results for the considered sample problems compare very well to analytical results.