Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Numerical solution of the high frequency asymptotic expansion for the scalar wave equation
Journal of Computational Physics
A PDE-based fast local level set method
Journal of Computational Physics
Weighted ENO Schemes for Hamilton--Jacobi Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
High-frequency wave propagation by the segment projection method
Journal of Computational Physics
A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods
SIAM Journal on Numerical Analysis
Geometric optics in a phase-space-based level set and Eulerian framework
Journal of Computational Physics
Nodal high-order methods on unstructured grids
Journal of Computational Physics
High-order RKDG Methods for Computational Electromagnetics
Journal of Scientific Computing
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
A Bloch band based level set method for computing the semiclassical limit of Schrödinger equations
Journal of Computational Physics
Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity
SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
The Chebyshev spectral viscosity method for the time dependent Eikonal equation
Mathematical and Computer Modelling: An International Journal
Journal of Computational Physics
Hi-index | 31.47 |
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.