Graduate Thesis Or Dissertation
 

Multiscale Methods and Energy Stable Discretizations for Maxwell’s Equations in Linearand Nonlinear Materials

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  • In this dissertation, we introduce a family of fully discrete finite difference time-domain (FDTD) methods for Maxwell’s equations in linear and nonlinear materials. Onecategory of methods is constructed using multiscale techniques involving operator splittings. We present the sequential splitting scheme, the Strang Marchuk splitting scheme,the weighted sequential splitting scheme including the symmetrical weighted sequentialsplitting scheme for the discretization of the time domain Maxwell’s equations in twoand three dimensions. We construct and analyze these operator splitting schemes basedon energy techniques for Maxwell’s equations in linear non-dispersive and non-dissipativematerials, and in nonlinear ferromagnetic materials. The fully discrete methods use theCrank-Nicolson scheme in time to enhance and improve stability and use staggering of electric and magnetic variables in space. As a consequence, we obtain fully discrete schemesthat are unconditionally stable. Moreover, we prove the convergence of solutions of ournew numerical techniques and provide comparisons with other relevant numerical methods, such as the Yee-FDTD scheme.The second category of methods involves efficient, accurate, and stable computational techniques, based on high order finite difference time domain (FDTD) methods inspace for Maxwell’s equations in a nonlinear optical medium. The nonlinearity comes from the instantaneous electronic Kerr response and the residual Raman molecular vibrational response, together with the single resonance linear Lorentz dispersion. We constructfully discrete modified second-order leap-frog and implicit trapezoidal temporal schemesto discretize the nonlinear terms in our Maxwell model. Under stability restrictions, thefully discrete modified leap-frog FDTD methods are proved to be stable under appropriate stability conditions, while the fully discrete trapezoidal FDTD methods are provedto be unconditionally stable. Finally, numerical experiments and examples are given thatillustrate and confirm our theoretical results.
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