A Semi-Lagrangian numerical method for Geometric Optics type problems
We explore an intermediate "Semi-Lagrangian" formulation of Geometric Optics type problems which stands between the classical "Lagrangian" Hamiltonian systems and its "Eulerian" Hamilton-Jacobi Partial Differential Equations counterpart. The goal is to design a numerical method which "trades" between advantages and drawbacks of both worlds to compute numerically the so-called "multi-valued" solution. We present a numerical algorithm and apply it to the "paraxial" simplification of 2-D geometric optics. We discuss its limitations and its possible extensions to higher dimensions. For an extended discussion on Lagrangian versus Eulerian methods in Geometric Optics and a review of some applications of these techniques see Engquist and Runborg, Acta Numer. (2003), and Benamou, J. Sci. Comput., 19 (2003).
geometric optic hamiltonian system eikonal equation hamilton-jacobi equation