An approximation scheme for an Eikonal Equation with discontinuous coefficient
This work provides a numerical method with error guarantees for a class of PDEs relevant to control and image processing, but the contribution is incremental as it extends existing semi-Lagrangian schemes to a specific type of discontinuity.
The authors propose a semi-Lagrangian scheme for numerically approximating viscosity solutions of stationary Hamilton-Jacobi equations with discontinuous coefficients, and prove an a-priori error estimate in an integral norm. Applications to control and image processing are demonstrated.
We consider the stationary Hamilton-Jacobi equation where the dynamics can vanish at some points, the cost function is strictly positive and is allowed to be discontinuous. More precisely, we consider special class of discontinuities for which the notion of viscosity solution is well-suited. We propose a semi-Lagrangian scheme for the numerical approximation of the viscosity solution in the sense of Ishii and we study its properties. We also prove an a-priori error estimate for the scheme in an integral norm. The last section contains some applications to control and image processing problems.