Abstract
A Schwarz Waveform Relaxation (SWR) algorithm is proposed to solve by Domain Decomposition Method (DDM) linear and nonlinear Schrödinger equations. The symbols of the transparent fractional transmission operators involved in Optimized Schwarz Waveform Relaxation (OSWR) algorithms are approximated by low order Lagrange polynomials to derive Lagrange–Schwarz Waveform Relaxation (LSWR) algorithms based on local transmission operators. The LSWR methods are numerically shown to be computationally efficient, leading to convergence rates almost similar to OSWR techniques.
Highlights
Introduction and methodologyLet us consider the following initial boundary-value problem: find the complex-valued wavefunction u(x, t) solution to the real-time NonLinear cubic Schrodinger Equation (NLSE) [2, 5, 8] set on Rd, d 1, i∂tu = − u + V (x)u + κ|u|2u, x ∈ Rd, t > 0, u(x, 0) = u0(x), x ∈ Rd, (1)with initial condition u0, nonlinearity strength κ 0 and smooth potential V (x) > 0
In 1d (d = 1), the general principle consists of approximating nonlocal fractional operators, such as ± −i∂t + V, for V constant, which are involved in Optimized Schwarz Waveform Relaxation (OSWR) transmission conditions [12] by simple low order partial differential operators
For space-dependent potentials V, OSWR methods for Linear Schrodinger Equation (LSE)/NLSE are derived from artificial operators ∂x + Λ±(x, ∂t), where Λ±(x, ∂t) = Op λ±(x, τ ), with λ±(x, τ ) = σ(Λ±) where τ is the covariable associated to t, through a Nirenberg factorization, and σ(A) is the symbol of a given pseudodifferential operator A
Summary
Despite the fast convergence of OSWR methods [10, 11, 12], their prohibitive cost in quantum wave problems is a consequence of the nonlocal character in time of the fractional operators Λ± and Λ±p [6, 7]. The interest of the presented methodology is three-fold: i) as the interpolation by Lagrange polynomials is performed from λ±p , we expect a fast SWR convergence [6], ii) as the corresponding transmission operators are local differential operators, we expect a competitive computational complexity compared to OSWR methods, and iii) the derivation of the Lagrange polynomial is performed from λ±p without additional assumption on the magnitude of |τ |.
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