High-Order Leap-Frog Based Discontinuous Galerkin Method for the Time-Domain Maxwell Equations on Non-Conforming Simplicial Meshes

Abstract

A high-order leap-frog based non-dissipative discontinuous Galerkin timedomain method for solving Maxwell's equations is introduced and analyzed. The proposed method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements, with a Nth-order leap-frog time scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary level hanging nodes. The method is proved to be stable under some CFL-like condition on the time step. The convergence of the semi-discrete approximation to Maxwell's equations is established rigorously and bounds on the global divergence error are provided. Numerical experiments with highorder elements show the potential of the method.

Analysis of a high-order space and time discontinuous Galerkin method for elastodynamic equations. Application to 3D wave propagation

“…Indeed, unlike the ADER method that has the same level of accuracy in space and time, the standard leap-frog time scheme, second-order accurate, reduces the global convergence order limiting the interest for high-order space discretizations. Then, we propose an extension of the leap-frog scheme to higher (even) orders of accuracy, following a method proposed for the Maxwell equations by Young [62] and applied in the DG context by Fahs [29]. This method allows us to achieve temporal accuracy to any even order desired, when free surface conditions are applied, by introducing an iterative procedure.…”

High-Order Time Discretization of The Wave Equation by Nabla-P Scheme.

“…Note that, for P 3 polynomials, we do not obtain the expected fourth order convergence. This is due to the fact that the original DG method does not converge with the optimal order [2,6,7]. To obtain the fourth order convergence, we need to use at least P 4 polynomials.…”

“…This scheme requires also three times more multiplications by the stiffness matrices than the LF scheme, but the stability condition is multiplied by 2.8 [6,7]. There is thus a offsetting effect.…”

POD-based model order reduction with an adaptive snapshot selection for a discontinuous Galerkin approximation of the time-domain Maxwell's equations

“…The resulting full DGTD method with LF 2 time scheme (14) is analyzed in [44] where it is shown that the method is non-dissipative, conserves a discrete form of the electromagnetic energy and is stable under the CFL condition…”