Alternating-Direction Implicit Formulation of the Finite-Element Time-Domain Method

Implementation of a First-Order ABC in Mixed Finite-Element Time-Domain Formulations Using Equivalent Currents

“…So, we can come to the conclusion that the LF-FETD and the CN-FETD are mixed versions of the VW-FETD discretized by the central-difference and the Newmark-with , respectively. This may not only justify stability of the CN-FETD for arbitrary large time steps in numerical experiments [17], but also help to decrease dissipation of the US hybrid FETD-FDTD [18]. Furthermore, it shows that our approach to implement the ABC in the mixed FETD formulations is valid and leads to the same result as the VW-FETD truncated with that ABC.…”

“…which is known as the Crank-Nicolson FETD (CN-FETD) [17]. Eliminating the magnetic field variable results in (16) Clearly, this equation is exactly (13) with .…”

A Further Study on the Use of the Alternating-Direction Implicit Scheme for the Finite-Element Time-Domain Method

“…The ADI formulation for time discretization [4], [14]- [19] is implemented for each time-step, and the following equations are derived:…”

“…Therefore, local matrixmatrix multiplication can be taken before assembling. Letting , and then assembling the local equations to a global system, the following equations are obtained: Compared to [4], neither global sparse matrix-matrix multiplication nor the matrix inverse is required to be computed before (25)-(28) are solved. Since the matrix in (25) and (27) is a positive definite, the LU factorization is taken only once for all the time iterations in a serial computer.…”

“…With the Newmark-beta scheme, the second-order wave equation FETD method is easily formulated to be unconditionally stable, and the time-step size is no longer limited by the space discretization. However, the mixed FETD had been limited to conditionally stable implementations until recently when an alternating-direction implicit (ADI) implementation was proposed in [4].…”