Krylov Subspace Approximation for TEM Simulation in the Time Domain

Forward transient electromagnetic modeling requires the numerical solution of a linear constant-coefficient initial-value problem for the quasi-static Maxwell equations. After discretization in space this problem reduces to a large system of ordinary differential equations, which is typically solved using finite-difference time-stepping. We compare standard time-stepping schemes such as the explicit and unconditionally stable Du Fort-Frankel scheme with the more recent Runge-Kutta-Chebyshev methods, which are designed specifically for parabolic initial value problems, with Krylov subspace techniques for the explicit solution of the initial value problem using the matrix exponential. Besides the classic Arnoldi/Lanczos approximation we also consider restarted Arnoldi approximations as were recently proposed in (Eiermann & Ernst, 2006). These restarted schemes have the advantage of requiring only an a priori fixed amount of memory storage, a significant aspect in the context of 3D simulations. We also present a recent efficient implementation (Afanasjew, Ernst, Güttel, & Eiermann, to appear) of the restarted Arnoldi method for evaluating the matrix exponential.