Energy stable discontinuous Galerkin methods for Maxwell's equations in nonlinear optical media

Implicit Runge-Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations

In this paper we consider implicit Runge–Kutta methods for the time-integration of linear Maxwell’s equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauß and Radau collocation methods. The results for the abstr...

A Discontinuous Galerkin Method for the Time-Domain Solution of 3D Maxwell's Equations on Non-Conforming Locally Refined Grids

A discontinuous Galerkin method is proposed for the numerical solution of the three-dimensional time-domain Maxwell's equations. A leap frog scheme is used for advancing in time. The scheme resulting can handle highly heterogeneous material, non diffusive and highly adaptable (it has been implemented on tetrahedral or hexahedral grids, including non-conforming). In some cases, some divergence c...

Optimized Schwarz method for solving time-harmonic Maxwell's equations discretized by a discontinuous Galerkin method

Local discontinuous Galerkin methods for nonlinear dispersive equations

We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions, the so-called ‘‘compactons’’. The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdV-type equations. We present two classes of DG metho...

Final Report of NASA Langley Grant NCC1-01035 Relaxation and Preconditioning for High Order Discontinuous Galerkin Methods with Applications to Aeroacoustics and High Speed Flows

methods are two classes of high order, high resolution methods suitable for convection dominated simulations with possible discontinuous or sharp gradient solutions. In [18], we first review these two classes of methods, pointing out their similarities and differences in algorithm formulation, theoretical properties, implementation issues, applicability, and relative advantages. We then present...