Numerical methods for Hamilton–Jacobi type equations

We give an overview of numerical methods for first-order Hamilton–Jacobi equations. After a short presentation of the theory of viscosity solutions, we show their link with entropy solutions of conservation laws. Then, we review theory and construction of monotone numerical methods in finite difference and semi-Lagrangian form, also providing a numerical test which shows the main features of this class of schemes. Finally, we sketch the main ideas behind high-order methods and more recent developments.