Artificial boundary conditions for certain evolution PDEs with cubic nonlinearity for non-compactly supported initial data

A friendly review of absorbing boundary conditions and perfectly matched layers for classical and relativistic quantum waves equations

“…We also assume that the initial wave function ψ 0 is compactly supported (that is ψ 0 is localized in space; some developments for non compactly supported initial data are available in [25,47]).…”

Frozen Gaussian approximation based domain decomposition methods for the linear Schrödinger equation beyond the semi-classical regime

“…Such conditions were developed for many kinds of wave equations. We refer here to some of analytical and numerical works, mainly based on pseudodifferential operators (although other robust techniques exist such as perfectly matched layers (PMLs) [16]): wave equations [19], diffusion equation [22,23], linear [1,5,7,8,9,33], nonlinear Schrödinger equations [3,6,10,30,40,42], Maxwell [2], Klein-Gordon and Dirac equations [13]. From the DDM point of view, and most particularly for OSWR-type methods, the use of TBC-based transmission conditions leads to fast converging algorithms (in a few fixed-point iterations).…”

Absorbing boundary conditions for relativistic quantum mechanics equations

“…Some original techniques that we will use were proposed by [3,4,5] in the framework of Maxwell's equations and linear and nonlinear Schrödinger equations. The general approach presented by Barucq et al is now become standard and is applied to several kinds of equations, such as evolution equations with cubic nonlinearity [30]. This paper is an instance of these approaches in the framework of relativistic quantum mechanics for laser-particle TDDE and 2 TDKGE.…”