High-order finite difference scheme for Maxwell's equations with complex interface jump conditions.

arXiv: Numerical Analysis(2018)

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We propose a high-order finite-difference time-domain (FDTD) scheme based on the correction function method (CFM) to treat interfaces with complex geometry. We present a system of PDEs coming from Maxwellu0027s equations and for which the solution corresponds to a function (the correction function) that can be used to correct the FDTD scheme in the vicinity of interfaces. We perform a perturbation analysis on the CFMu0027s system of PDEs. A functional that is a square measure of the error satisfying the CFMu0027s system of PDEs is minimized in a divergence-free discrete functional space. This allows us to compute an approximation of the correction function where it is needed. The approximation of the correction function is then used to correct either a second-order or fourth-order FDTD scheme. We use a staggered grid in space to enforce the discrete divergence constraints and the fourth-order Runge-Kutta time-stepping method. The discrete divergence constraint, the stability and the error of the resulting scheme are studied. Numerical experiments are performed for problems with different geometries of the interface, including one with cusps. A second-order convergence is obtained for a second-order FDTD scheme with the CFM. High-order convergence is obtained with a fourth-order FDTD scheme with the CFM. The discontinuities within the solutions are accurately captured without spurious oscillations even around cusps.
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