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. One category of schemes is constructed using multiscale techniques involving operator splitting. We present the sequential splitting scheme, the Strang Marchuk splitting scheme, the weighted sequential splitting scheme including the symmetrical weighted sequential splitting scheme for the discretization of the time domain Maxwell's equations in two and three dimensions. We construct and analyze these operator splitting schemes based on energy techniques for Maxwell's equations in linear non-dispersive and non-dissipative materials, and in nonlinear ferromagnetic materials. The fully discrete methods use the Crank-Nicolson scheme in time to enhance and improve stability and use staggering of electric and magnetic variables in space as in the Yee-FDTD method. As a consequence, we obtain fully discrete schemes that are unconditionally stable. Moreover, we prove the convergence of solutions of our schemes and provide comparisons with other relevant numerical schemes, such as the Yee-FDTD method.
The second category of scheme involves efficient, accurate, and stable computational techniques, based on high order finite difference time domain (FDTD) methods in space 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 construct fully discrete modified second-order leap-frog and implicit trapezoidal temporal schemes to discretize the nonlinear terms in our Maxwell model. Under stability restrictions, the fully discrete modified leap-frog FDTD methods are proved to be stable under appropriate stability conditions, while the fully discrete trapezoidal FDTD methods are proved to be unconditionally stable. Finally, numerical experiments and examples are given that illustrate and confirm our theoretical results.
Puttha's major professor is Prof. Vrushali Bokil.