Conservative space-time mesh refinement methods for the FDTD solution of Maxwell’s equations

Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems

“…A simplified version will be presented in the next section too. A very elegant solution with Lagrange multipliers has been proposed [7] and enhanced [1], which leads however to the solution of a fixed linear system at each iteration. Another totally explicit solution was built, which seems to be only consistent in average and then first-order accurate [25].…”

“…For example, for the transient solution of hyperbolic conservation laws where several algorithms have been proposed [8,26], it is not easy to verify a maximum principle and to maintain high order accuracy. In our context, where a non-dissipative spatial discretization is used in order to achieve exact energy conservation, efficient local time-stepping is also difficult to built: a solution for wave propagation with Lagrange multipliers has been proposed [7] and enhanced [1], which leads however to the solution of a fixed, interface-sized linear system at each time iteration. Another totally explicit solution was built, which seems to be only consistent in average and then first-order accurate [25].…”

Multi-level explicit local time-stepping methods for second-order wave equations

“…In [13] Collino, Fouquet and Joly proposed an LTS method for the wave equation in first-order form, which conserves a discrete energy yet requires every time-step the solution of a linear system on the interface between the coarse and the fine mesh. It was analyzed in [14,15] and later extended to elastodynamics [16] and Maxwell's equations [17]. By combining a symplectic integrator with a DG discretization of Maxwell's equations in first-order form, Piperno [18] proposed a second-order explicit local time-stepping scheme, which also conserves a discrete energy.…”

Implicit Local Refinement for Evanescent Layers Combined With Classical FDTD

“…If no analytical solution is known, local refinement [4] seems like an attractive option were it not that this usually requires a significantly reduced time step (though several "multirate" refinement techniques have been proposed [5], [6] and appendix and several references in [4]). …”