An accurate spectral/discontinuous finite-element formulation of a phase-space-based level set approach to geometrical optics

Abstract

In this paper, we introduce a new numerical procedure for simulations in geometrical optics that, based on the recent development of Eulerian phase-space formulations of the model, can deliver very accurate, uniformly resolved solutions which can be made to converge with arbitrarily high orders in general geometrical configurations. Following previous treatments, the scheme is based on the evolution of a wavefront in phase-space, defined as the intersection of level sets satisfying the relevant Liouville equation. In contrast with previous work, however, our numerical approximation is specifically designed: (i) to take full advantage of the smoothness of solutions; (ii) to facilitate the treatment of scattering obstacles, all while retaining high-order convergence characteristics. Indeed, to incorporate the full regularity of solutions that results from the unfolding of singularities, our method is based on their spectral representation; to enable a simple high-order treatment of scattering boundaries, on the other hand, we resort to a discontinuous Galerkin finite element method for the solution of the resulting system of equations. The procedure is complemented with the use of a recently derived strong stability preserving Runge–Kutta (SSP-RK) scheme for the time integration that, as we demonstrate, allows for overall approximations that are rapidly convergent.