In this talk we will consider three different numerical methods for solving nonlinear PDEs:
- A class of Godunov-type second order schemes for nonlinear conservation laws, starting from the Nessyahu-Tadmor scheme;
- A class of L1 -based minimization methods for solving linear transport equations and stationary Hamilton- Jacobi equations;
- Entropy-viscosity methods for nonlinear conservation laws.
All of the above methods are based on high-order approximations of the corresponding nonlinear PDE and respect a weak form of an entropy condition. Theoretical results and numerical examples for the performance of each of the three methods will be presented.