An asymptotic Green's function method for the wave equation

Numerical solutions of the time‐dependent Schrödinger equation with position‐dependent effective mass

“…For evaluating the integrals ( 18) and ( 20) efficiently, we will explore the low-rank properties of (x, 𝜉) and apply the randomized pivoted QR factorization [24,53] to compute the low-rank approximations; also refer to Appendix. For a generic matrix (x, 𝜉), the desired low-rank approximation is given as…”

Green’s Function Methods: Some developments and applications

“…Solving equations (3.18) and (3.19) with mesh-based numerical methods is computationally too expensive, since they must be solved at each time step t n for each point in the Fourier space k. Therefore, we will seek analytic solutions in terms of Taylor series expansions of the solutions for a short period of time ; ; Mayfield et al (2021). We assume the phase and amplitude terms have Taylor series expansions centered at each time step t n with the form, (3.21) and (3.22) for t = t n + ∆t, with the expansion terms {τ ℓ (x, k)} and {A mℓ (x, k)} to be determined.…”

“…The wave is split into its forward-propagating and backward-propagating parts that can be propagated by an efficient time propagator in the form of a pseudo-differential operator, along with lowrank approximations ; Goreinov et al (1997b,a) and fast Fourier transforms (FFT) . This method was extended and improved in Mayfield et al (2021) by designing the time propagator, namely the asymptotic Green's function method (AGFM), in the form of an integral with Green's function based on Huygens' principle, where the Green's function is approximated asymptotically by geometrical optics approximations ; ; ; ; ; . Along with lowrank approximations ; Goreinov et al (1997b,a) and fast Fourier transforms (FFT) , the approximated integral can be computed efficiently to provide more accurate numerical solutions.…”

“…Here, we will combine the ideas in ; ; ; Mayfield et al (2021) and ; to develop an efficient time propagator for the vector wave equations (3.1), aiming to balance efficiency and accuracy. The problem will be firstly restricted to a bounded domain of interest by applying the perfectly matched layer (PML) method ; that surrounds the bounded domain by an absorbing layer such that the information of the waves outside of the bounded domain becomes unimportant.…”

“…Following the PML method, the WKBJ approximations, the analytic approximations for the phase and amplitudes of the asymptotic dyadic Green's function, and the lowrank approximations through a randomized pivoted QR factorization are combined to derive an efficient short-time propagator for propagating the forward-propagating and backward-propagating waves governed by equations (3.9). The method, as the asymptotic Green's function method in Mayfield et al (2021), has complexity O(N log N ) per time step with the pre-factor dependent of the low rank for a prescribed accuracy requirement. With the one-term WKBJ approximation for the dyadic Green's function and second-order accurate short-time Taylor expansions for its phase and amplitude, the method is first-order accurate.…”