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Asymptotic Convergence Rates of Schwarz Waveform Relaxation Algorithms for Schrödinger Equations with an Arbitrary Number of Subdomains

  • Xavier Antoine
  • Emmanuel Lorin
Original Research
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Abstract

We derive some asymptotic estimates of the rate of convergence of Schwarz Waveform Relaxation domain decomposition methods for the Schrödinger equation when using an arbitrary number of subdomains. Hence, we justify that under certain conditions, the rates of convergence mathematically obtained for two subdomains (Antoine et al. in ESAIM M2AN,  https://doi.org/10.1051/m2an/2017048, 2018; Antoine and Lorin in Numer Math 137(4):923–958, 2017; Antoine et al. in (submitted), 2018) are still asymptotically valid for a larger number of subdomains, as it is usually numerically observed (Halpern and Szeftel in Math Models Methods Appl Sci 20(12):2167–2199, 2010).

Keywords

Schwarz Waveform Relaxation Domain decomposition methods Schrödinger equation 

Introduction

We are interested in this paper in the analysis of the rate of convergence of some SWR Domain Decomposition Methods (DDMs) by using an arbitrary number of subdomains. This study is an extension of existing results about the convergence of SWR algorithms on two subdomains [8, 9, 10]. We show that the convergence rates established for two subdomains are actually still accurate estimates for an arbitrary number of sufficiently large subdomains and bounded potentials. In this paper, we will mainly focus on the computation of contraction factors from Lipschitz continuous mappings involved in the proof of convergence of SWR methods. As a consequence, we will not introduce technical details about the full proof of convergence. Instead, we refer to several papers where the reader could find them, depending on the equation under consideration. For linear advection and diffusion reaction equations, we refer to [18]. The analysis and derivation for the Schrödinger equation in the time-dependent case is presented in [10, 11, 16, 17], while the stationary equation is studied in [8, 9].

The SWR algorithms are well-established methods for the parallel solution of evolution partial differential equations by allowing computations of the underlying PDE on small subdomains with very good speed-up. The convergence of the overall SWR DDM is strongly dependent on the choice of the transmission conditions between the subdomain interfaces. Typically, classical Dirichlet transmission conditions will usually provide very slow convergence, while transparent transmission conditions provide a very fast converging solution. In this paper, we do not discuss the space discretization of the algorithm, but focus on the convergence of the continuous in-space algorithms. We refer to [1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] for details about the SWR methods. In this paper, we analyze the convergence rate for the Classical SWR (CSWR) algorithm which is based on Dirichlet transmission conditions, and Optimal SWR algorithm (OSWR) based on transparent transmission conditions. In the latter case, convergence of the 2-subdomain algorithm is optimal in the sense that it occurs in only 2 iterations [9, 24].

This paper is organized as follows. In Sect. 2, we analyze the rate of convergence of the CSWR and OSWR DDMs for the stationary Schrödinger equation solved by the imaginary-time method. The analysis is extended in Sect. 3 to real-time dynamics, and some numerical experiments are proposed in Sect. 4.

Stationary States Problems

We study the convergence of the SWR methods by using an arbitrary number \(m \geq 2\) of subdomains, for computing the point spectrum of the Schrödinger Hamiltonian by using the imaginary-time method [3, 5, 6, 7, 12, 13, 14, 15]. We refer to [8, 18] regarding the well-posedness of the equation and of the convergence of the algorithms. We intend to determine the ground state to the following one-dimensional Schrödinger Hamitonian \(-\triangle +V(x)\) (used in quantum mechanics, acoustics wave propagation, optics), where the potential V is supposed to be smooth and bounded with bounded derivative. The imaginary-time method, which is also called Normalized Gradient Flow (NGF) method, reads: for \(t_0=0< t_1< \cdots< t_n<t_{n+1} <\cdots\), solve
$$\begin{aligned} \left\{ \begin{array}{l} \partial _t \phi (t,x) = \triangle \phi (t,x) - V(x)\phi (t,x) \phi (t,x), \, x \in \varOmega ,\, t_{n}< t<t_{n+1}, \\ \\ \phi (t,x) = 0, \, x \in \partial \varOmega , \, t_{n}< t <t_{n+1},\\ \\ \phi (t_{n+1},x):=\phi (,t^{+}_{n+1},x) =\frac{\phi (t^{-}_{n+1},x)}{|| \phi (\cdot ,t^{-}_{n+1})||_{L^{2}(\mathbb {R})}},\\ \\ \phi (0,x)=\varphi _0(x), \, x \in \mathbb {R}, \text {with }||\varphi _{0} ||^{2}_{L^{2}(\mathbb {R})}=1, \end{array} \right. \end{aligned}$$
(1)
where \(\varphi _{0}\) is a given initial guess, usually built from an ansatz. The procedure is repeated until the convergence is reached, that is when the following stopping criterion is satisfied
$$\begin{aligned} || \phi (t_{n+1},\cdot ) - \phi (t_{n},\cdot )||_{L^{\infty }(\mathbb {R})}\leq \delta , \end{aligned}$$
(2)
for \(\delta >0\) small enough. We propose the following decomposition (with possible overlap): \(\varOmega = \cup _{i=1}^m\varOmega _i\), such that \(\varOmega _i = (\xi _i^{-},\xi _i^{+})\), for \(2\leq i \leq m-1\), \(\varOmega _1=(-a,\xi _1^{+})\) and \(\varOmega _m=(\xi _m^{-},+a)\) (see Fig. 1). Moreover, the overlapping size is: \(\xi _{i}^{+}-\xi _{i+1}^{-} = \varepsilon >0\), and \(|\varOmega _i| = L+\varepsilon\), where L is assumed to be much larger than \(\varepsilon\). We also have \(\xi _{i+1}^{\pm }-\xi _i^{\pm } = L\). In the following, we will denote by \(\phi ^{(k)}_i\) the local solution in the subdomain \(\varOmega _i\) at Schwarz iteration k.
Fig. 1

Domain decomposition with possible overlapping

We study the convergence rate of the CSWR and OSWR algorithms. The convergence rate is here defined as the slope of the logarithm residual history with respect to the Schwarz iteration number, i.e. \(\{(k,\log (E^{(k)})) \, : \, k \in \mathbb {N}\}\), where
$$\begin{aligned} E^{(k)}:& = \sum _{i=1}^{m-1}\big \Vert \ \ \Vert \phi ^{\text {cvg},(k)}_{i\big |(\xi _{i+1}^-,\xi _i^+)}\nonumber \\&-\phi ^{\text {cvg},(k)}_{i+1\big |(\xi _{i+1}^-,\xi _i^+)}\Vert _{\infty }\big \Vert _{L^{2}(0,T^{(k^{\text {cvg}})})}\leq \delta ^{\text {Sc}}, \end{aligned}$$
(3)
and \(\phi _{i}^{\text {cvg},(k)}\) (resp. \(T^{(k^{\text {cvg}})}\)) denotes the NGF converged solution (resp. time) in \(\varOmega _i\) at Schwarz iteration k, \(\delta ^{\text {Sc}}\) being a small parameter. More specifically, we intend to prove that the convergence rate is, in first approximation, independent of the number of subdomains as already numerically mentioned in [24] and Sect. 4. Finally, \(\phi ^{(k)}_i\) is the local solution in \(\varOmega _i\), for any \(i \in \{1,\cdots ,m\}\), at Schwarz iteration \(k \in \mathbb {N}\).

Potential-Free Equation

We first consider the potential-free Schrödinger equation in imaginary-time, with \(P(\partial _t,\partial _x) = \partial _t - \partial ^2_{x}\). The NGF algorithm consists in solving for any \(n \in \mathbb {N}\), from \(t_n\) to \(t_{n+1}^-\):
$$\begin{aligned} \left\{ \begin{array}{lll} \big (\partial _t - \partial ^2_{x}\big ) \phi (t,x) & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega , \\ \phi (t,-a) & = 0, \qquad t \in (t_n,t_{n+1}^-),\\ \phi (t,a) & = 0, \qquad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
We then \(L^{2}(\mathbb {R})\)-normalize the solution
$$\begin{aligned} \phi (t_{n+1},\cdot ) = \frac{\phi (t_{n+1^-},\cdot )}{\Vert \phi (t_{n+1^-})\Vert _{L^2(\varOmega )}}. \end{aligned}$$
The procedure is repeated until convergence following the stopping criterion (2).

Let us start by studying the rate of convergence of the CSWR, then considering the OSWR algorithm. We will first generalize the ideas and results presented in [18] to m subdomains, as well as [9].

CSWR Algorithm

The CSWR algorithm reads as follows: for \(k \geq 1\) and \(i \in \{2,\ldots ,m-1\}\), from time \(t_n\) to \(t_{n+1}^-\), solve
$$\begin{aligned} \left\{ \begin{array}{lll} P(\partial _t,\partial _x)\phi _i^{(k)} & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _i,\\ \phi _i^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _i, \\ \phi ^{(k)}_i(t,\xi _i^{+}) & = \phi ^{(k-1)}_{i+1}(t,\xi _{i}^{+}), \qquad t \in (t_n,t_{n+1}^-), \\ \phi ^{(k)}_i(t,\xi _i^{-}) & = \phi ^{(k-1)}_{i-1}(t,\xi _{i}^{-}), \qquad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
In \(\varOmega _1\), we get
$$\begin{aligned} \left\{ \begin{array}{lll} P(\partial _t,\partial _x)\phi _1^{(k)} & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _1,\\ \phi _1^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _1, \\ \phi ^{(k)}_1(t,-a) & = 0, \qquad t \in (t_n,t_{n+1}^-), \\ \phi ^{(k)}_1(t,\xi _1^{+}) & = \phi ^{(k-1)}_{2}(t,\xi _{2}^{+}), \qquad t \in (t_n,t_{n+1}^-), \end{array} \right. \end{aligned}$$
and in \(\varOmega _m\)
$$\begin{aligned} \left\{ \begin{array}{lll} P(\partial _t,\partial _x)\phi _m^{(k)} & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _m,\\ \phi _m^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _m, \\ \phi ^{(k)}_i(t,\xi _m^{-}) & = \phi ^{(k-1)}_{m-1}(t,\xi _{m}^{-}), \qquad t \in (t_n,t_{n+1}^-), \\ \phi ^{(k)}_m(t,+a) & = 0, \qquad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
To analyze the convergence of this DDM, we set \(e_i := \phi _{\text {exact}|\varOmega _i} - \phi _i\) and we introduce \(h_i^{\pm } \in H_0^{3/4}(0,T)=\big \{\phi \in H^{3/4}(0,T) \, : \, \phi (0)=0\big \}\), for \(i \in \{1,\ldots ,m\}\). We consider the following system
$$\begin{aligned} \left\{ \begin{array}{lll} \big (\mathtt{i}\tau - \partial _x^2\big ){\widehat{e}}_i(\tau ,x) & = 0, \qquad (\tau ,x) \in \mathbb {R}\times \varOmega _i, \\ {\widehat{e}}_i(\tau ,\xi ^{\pm }) & = {\widehat{h}}_i^{\pm }(\tau ), \qquad \tau \in \mathbb {R}\, , \end{array} \right. \end{aligned}$$
where we denote by \(\tau\) the dual variable to t in Fourier space, and by \(\, {\widehat{\cdot }} \,\) the Fourier transform \(\mathcal {F}\) with respect to t. We set \(\alpha (\tau ) := e^{\mathtt{i}\pi /4}\sqrt{\tau }\), and we get \({\widehat{e}}_i(\tau ,x) = A_i(\tau )e^{\alpha (\tau )x} + B_i(\tau )e^{-\alpha (\tau )x}\), for any \(\tau \in \mathbb {R}\), and for \(i \in \{2,\ldots ,m-1\}\)
$$\begin{aligned} {\widehat{e}}_i(\tau ,\xi _i^{\pm })& = A_i(\tau )e^{\alpha (\tau )\xi _i^{\pm }} + B_i(\tau )e^{\alpha (\tau )\xi _i^{\pm }}\\& = h_i^{\pm }(\tau ) \,. \end{aligned}$$
By using the boundary / transmission conditions, we find, for \(i \in \{2,\ldots ,m-1\}\),
$$\begin{aligned} \left\{ \begin{array}{c} \displaystyle A_i(\tau ) = \frac{\big ({\widehat{h}}_i^+(\tau )e^{-\alpha (\tau )\xi _i^-} - {\widehat{h}}_i^-(\tau )e^{-\alpha (\tau )\xi _i^+}\big )}{e^{\alpha (\tau )(L+\varepsilon )} - e^{-\alpha (\tau )(L+\varepsilon )}}, \\ \displaystyle B_i(\tau ) = \frac{\big ({\widehat{h}}_i^-(\tau )e^{\alpha (\tau )\xi _i^-} - {\widehat{h}}_i^+(\tau )e^{\alpha (\tau )\xi _i^+}\big )}{e^{\alpha (\tau )(L+\varepsilon )} - e^{-\alpha (\tau )(L+\varepsilon )}}. \end{array} \right. \end{aligned}$$
Similarly, we have
$$\begin{aligned} \left\{ \begin{array}{c} \displaystyle A_1(\tau ) = \frac{{\widehat{h}}_1^+(\tau )}{e^{\alpha (\tau )(L+\varepsilon /2-a)} - e^{-\alpha (\tau )(L+\varepsilon /2+a)}} \, , \\ \displaystyle B_1(\tau ) = \frac{{\widehat{h}}_1^+(\tau )}{e^{\alpha (\tau )(L+\varepsilon /2+a)} - e^{-\alpha (\tau )(L+\varepsilon /2-a)}} \, , \end{array} \right. \end{aligned}$$
and
$$\begin{aligned} \left\{ \begin{array}{c} \displaystyle A_m(\tau ) = \frac{{\widehat{h}}_m^-(\tau )}{e^{\alpha (\tau )(a-L-\varepsilon /2)} - e^{\alpha (\tau )(L+\varepsilon /2+a)}} \, ,\\ \displaystyle B_m(\tau ) = -\frac{{\widehat{h}}_m^-(\tau )}{e^{-\alpha (\tau )(L+\varepsilon /2+a)}-e^{\alpha (\tau )(L+\varepsilon /2-a)}} \, . \end{array} \right. \end{aligned}$$
We introduce the mapping \(\mathcal {G}^{(C)}\) from \(\big (H^{3/4}(\mathbb {R})\big )^{2(m-1)}\) to itself, defined as follows: \(\mathcal {G}^{(C)}: \langle \big \{h_i^{+},h^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \rangle = \langle \big \{e_i(\cdot ,\xi _{i+1}^-),e_{i+1}(\cdot ,\xi _{i}^{+}) \big \}_{1 \leq i \leq m-1}\rangle \,\). Thus, we deduce that
$$\begin{aligned}&\mathcal {F}\big (\mathcal {G}^{(C)} \left\langle \big \{h_i^{+},h^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \right\rangle \big )(\tau )\\&\quad = \left\langle \big \{{\widehat{e}}_i(\tau ,\xi _{i+1}^-),{\widehat{e}}_{i+1}(\tau ,\xi _{i}^{+})\big \}_{1 \leq i \leq m-1} \right\rangle , \end{aligned}$$
where
$$\begin{aligned} \displaystyle {\widehat{e}}_i(\tau ,\xi _{i+1}^-) = \frac{\big ({\widehat{h}}_i^-(\tau )(e^{\varepsilon \alpha (\tau )} - e^{-\varepsilon \alpha (\tau )}) + {\widehat{h}}_i^+(\tau )(e^{L\alpha (\tau )} - e^{-L\alpha (\tau )}) \big )}{e^{\alpha (\tau )(L+\varepsilon )} - e^{-\alpha (\tau )(L+\varepsilon )}} \end{aligned}$$
and
$$\begin{aligned} \displaystyle {\widehat{e}}_{i+1}(\tau ,\xi _{i}^+) = \frac{\big ({\widehat{h}}_{i+1}^+(\tau )(e^{\varepsilon \alpha (\tau )} - e^{-\varepsilon \alpha (\tau )}) + {\widehat{h}}_{i+1}^-(\tau )(e^{L\alpha (\tau )} - e^{-L\alpha (\tau )}) \big )}{e^{\alpha (\tau )(L+\varepsilon )} - e^{-\alpha (\tau )(L+\varepsilon )}} \, . \end{aligned}$$
We then introduce
$$\begin{aligned} \widehat{\mathcal {G}}^{(C)}: \left\langle \big \{{\widehat{h}}_i^{+},{\widehat{h}}^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \right\rangle = \left\langle \big \{{\widehat{e}}_i(\cdot ,\xi _{i+1}^-),{\widehat{e}}_{i+1}(\cdot ,\xi _{i}^{+})\big \}_{1 \leq i \leq m-1} \right\rangle , \end{aligned}$$
Following the same strategy as for the two subdomains case [9, 24], we compute \(\widehat{\mathcal {G}}^{(C),2}\big (\langle \{{\widehat{h}}^+_i,{\widehat{h}}_{i+1}^-\}_{i}\rangle \big )\). Let us set \({\widehat{h}}_i^{+,(2)}(\tau ) := {\widehat{e}}_i(\tau ,\xi _{i+1}^-)\) and \({\widehat{h}}_{i+1}^{-,(2)}(\tau ) := {\widehat{e}}_{i+1}(\tau ,\xi _{i}^+)\). We get
$$\begin{aligned}&\displaystyle {\widehat{e}}_i^{(2)}(\tau ,\xi _{i+1}^-) \\&\quad = \frac{\Big ({\widehat{e}}_i(\tau ,\xi _{i+1}^-)\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big ) + {\widehat{e}}_i(\tau ,\xi _{i-1}^+)\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )\Big )}{e^{\alpha (\tau )(L+\varepsilon )}-e^{-\alpha (\tau )(L+\varepsilon )}} \displaystyle \end{aligned}$$
and
$$\begin{aligned}&\displaystyle {\widehat{e}}_{i+1}^{(2)}(\tau ,\xi _{i}^+) \\&\quad = \frac{\Big ({\widehat{e}}_{i+1}(\tau ,\xi _{i}^+)\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big ) + {\widehat{e}}_{i+1}(\tau ,\xi _{i+2}^-)\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )\Big )}{e^{\alpha (\tau )(L+\varepsilon )}-e^{-\alpha (\tau )(L+\varepsilon )}}. \end{aligned}$$
We then obtain
$$\begin{aligned} \displaystyle&{\widehat{e}}_{i+1}^{(2)}(\tau ,\xi _{i}^+) = \frac{1}{\big (e^{\alpha (\tau )(L+\varepsilon )}-e^{-\alpha (\tau )(L+\varepsilon )}\big )^2}\\&\qquad \Big \{{\widehat{h}}^{-}_{i+1}(\tau )\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big )^2 + {\widehat{h}}^{-}_{i+1}(\tau )\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )^2 \\&\qquad + {\widehat{h}}^{-}_{i+1}(\tau )\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big )^2 \\&\qquad + 2{\widehat{h}}^{+}_{i+1}(\tau )\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big ) \Big \}\\&\quad = \frac{e^{-2\varepsilon \alpha (\tau )}}{1-e^{-2\alpha (\tau )(L+\varepsilon )}\big )^2}\Big \{{\widehat{h}}^{-}_{i+1}(\tau )\big (1-e^{-2\alpha (\tau )L}\big )^2\\&\qquad + {\widehat{h}}^{-}_{i+1}(\tau )\big (e^{\alpha (\tau )(\varepsilon -2L)}-e^{-\alpha (\tau )(\varepsilon -2L)}\big )^2 \\&\qquad + 2{\widehat{h}}^{+}_{i+1}(\tau )\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )\big (e^{-\alpha (\tau )L}-e^{-3\alpha (\tau )L}\big ) \Big \} \, \end{aligned}$$
and
$$\begin{aligned}&{\widehat{e}}_{i}^{(2)}(\tau ,\xi _{i+1}^-) = \frac{1}{\big (e^{\alpha (\tau )(L+\varepsilon )}-e^{-\alpha (\tau )(L+\varepsilon )}\big )^2}\\&\qquad \Big \{{\widehat{h}}^{+}_{i}(\tau )\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big )^2 + {\widehat{h}}^{+}_{i}(\tau )\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )^2 \\&\qquad + 2{\widehat{h}}^{-}_{i}(\tau )\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )\big (e^{\alpha (\tau )L}-e^{-\alpha (\tau )L}\big ) \Big \} \\&\quad = \frac{e^{-2\alpha (\tau )\varepsilon }}{\big (1-e^{-2\alpha (\tau )(L+\varepsilon )}\big )^2}\Big \{{\widehat{h}}^{+}_{i}(\tau )\big (1-e^{-2\alpha (\tau )L}\big )^2\\&\qquad + {\widehat{h}}^{+}_{i}(\tau )\big (e^{\alpha (\tau )(\varepsilon -2L)}-e^{-\alpha (\tau )(\varepsilon -2L)}\big )^2 \\&\qquad + 2{\widehat{h}}^{-}_{i}(\tau )\big (e^{\alpha (\tau )\varepsilon }-e^{-\alpha (\tau )\varepsilon }\big )\big (e^{-\alpha (\tau )L}-e^{-3\alpha (\tau )L}\big ) \Big \} \, . \end{aligned}$$
For \(m \geq 2\) subdomains, we have
$$\begin{aligned}&\widehat{\mathcal {G}}^{(C),2}\big (\langle \{{\widehat{h}}^+_i,{\widehat{h}}_{i+1}^-\}_{i}\rangle \big )(\tau )\\&\quad = e^{-2\alpha (\tau )\varepsilon }\left\langle \big \{{\widehat{h}}_i^+(\tau ), {\widehat{h}}_{i+1}^-(\tau )\big \}_{i}\right\rangle + O\big (e^{-\alpha (\tau )L}\big ) \, . \end{aligned}$$
In particular, for \(\tau <0\), we obtain
$$\begin{aligned}&\big |\widehat{\mathcal {G}}^{(C),2}\big (\langle \{{\widehat{h}}^+_i,{\widehat{h}}_{i+1}^-\}_{i} \rangle \big )(\tau )\big |\\&\quad \leq e^{-\varepsilon \sqrt{2|\tau |}}\big |\left\langle \big \{{\widehat{h}}_i^+(\tau ), {\widehat{h}}_{i+1}^-(\tau ) \big \}_i\right\rangle \big | + Ce^{-L\sqrt{2|\tau |}}. \end{aligned}$$
Following exactly the same procedure as [18] or [9], we claim for any \(n \in \mathbb {N}^*\)
$$\begin{aligned}&\big \Vert \left\langle \{ e^{2k+1}_i ,e^{2k+1}_{i+1}\}_{1 \leq i \leq m-1}\right\rangle \big \Vert _{\Pi _{i=1}^{m-1}H^{2,1}(\varOmega _{i} \times (0,{\widetilde{T}}))\times H^{2,1}(\varOmega _{i+1} \times (0,{\widetilde{T}}))} \\&\quad \leq (C_{\varepsilon }^{(C)})^k \times \Vert \left\langle \big \{ e_i^{(0)}(\xi _i^+),e_{i+1}^{(0)}(\xi _{i+1}^- \big \}_{1 \leq i \leq m-1} \right\rangle \Vert _{\big (H^{3/4}(0,{\widetilde{T}})\big )^{2(m-1)}} \end{aligned}$$
for some \({\widetilde{T}}\), with \(C_{\varepsilon }^{(C)}\) a positive constant lower that 1. This justifies the fact that, as numerically observed in [24] and in Sect. 4, the convergence rate is independent of the number of subdomains of length L. However, the overall convergence is linearly dependent of the number of subdomains through the summation over the m subdomains in (3). As a consequence, we expect that the logscale slope of the residual history, i.e. the convergence rate, to be independent of m. Finally, the overall error \(\{(k,E^{(k)})\}\) is still shifted in logscale, by a positive constant linearly dependent on \(\log (m)\).

We conclude by the following

Proposition 1

The convergence rate \(C_{\varepsilon }^{(\text {C})}\) of the CSWR method with subdomains of length L is of the form
$$\begin{aligned} C_{\varepsilon }^{(\text {C})}& = \sup _{\tau } e^{-\varepsilon \sqrt{2|\tau |}} + O\big (e^{-L\sqrt{2|\tau |}}\big ). \end{aligned}$$

OSWR Algorithm

We now study the OSWR algorithm for \(m \geq 2\) subdomains. To this end, we define the transparent boundary operator \(\partial _x\pm \partial _t^{1/2}\). The OSWR algorithm reads as follows: for \(k \geq 1\) and for \(i \in \{1,\ldots ,m-1\}\), solve
$$\begin{aligned} \left\{ \begin{array}{lll} P(\partial _t,\partial _x)\phi _i^{(k)} & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _i,\\ \phi _i^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _i, \\ (\partial _x+\partial _t^{1/2})\phi ^{(k)}_i(t,\xi _i^{+}) & = (\partial _x+\partial _t^{1/2})\phi ^{(k-1)}_{i+1}(t,\xi _{i}^{+}), \qquad t \in (t_n,t_{n+1}^-), \\ (\partial _x-\partial _t^{1/2})\phi ^{(k)}_i(t,\xi _i^{-}) & = (\partial _x-\partial _t^{1/2})\phi ^{(k-1)}_{i-1}(t,\xi _{i}^{-}), \qquad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
In \(\varOmega _1\), we get
$$\begin{aligned} \left\{ \begin{array}{lll} P(\partial _t,\partial _x)\phi _1^{(k)} & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _1,\\ \phi _1^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _1, \\ \phi ^{(k)}_1(t,-a) & = 0, \qquad t \in (t_n,t_{n+1}^-), \\ (\partial _x+\partial _t^{1/2})\phi ^{(k)}_1(t,\xi _1^{+}) & = (\partial _x+\partial _t^{1/2})\phi ^{(k-1)}_{2}(t,\xi _{1}^{+}), \qquad t \in (t_n,t_{n+1}^-), \end{array} \right. \end{aligned}$$
and in \(\varOmega _m\)
$$\begin{aligned} \left\{ \begin{array}{lll} P(\partial _t,\partial _x)\phi _m^{(k)} & = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _m,\\ \phi _m^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _m, \\ (\partial _x-\partial _t^{1/2})\phi ^{(k)}_i(t,\xi _m^{-}) & = (\partial _x-\partial _t^{1/2})\phi ^{(k-1)}_{m-1}(t,\xi _{m}^{-}), \qquad t \in (t_n,t_{n+1}^-), \\ \phi ^{(k)}_m(t,+a) & = 0, \qquad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
We then consider
$$\begin{aligned} \left\{ \begin{array}{lll} \big (\mathtt{i}\tau - \partial _x^2\big ){\widehat{e}}_i(\tau ,x) & = 0, \qquad (\tau ,x) \in \mathbb {R}\times \varOmega _i, \\ (\partial _x \pm \alpha (\tau )){\widehat{e}}_i(\tau ,\xi ^{\pm }) & = {\widehat{h}}_i^{\pm }(\tau ), \qquad \tau \in \mathbb {R}\, . \end{array} \right. \end{aligned}$$
By defining \(\alpha (\tau ) := e^{\mathtt{i}\pi /4}\sqrt{\tau }\), we obtain
$$\begin{aligned} {\widehat{e}}_i(\tau ,x)& = A_i(\tau )e^{\alpha (\tau )x} + B_i(\tau )e^{-\alpha (\tau )x} \, . \end{aligned}$$
By again using the boundary conditions, we find, for \(i \in \{2,\ldots ,m-1\}\),
$$\begin{aligned} A_i(\tau ) = \frac{{\widehat{h}}_i^+(\tau )e^{-\alpha (\tau )\xi _{i}^+}}{2\alpha (\tau )}, \qquad B_i(\tau ) = -\frac{{\widehat{h}}_i^-(\tau )e^{\alpha (\tau )\xi _{i}^-}}{2\alpha (\tau )} \, . \end{aligned}$$
We introduce a mapping \(\mathcal {G}^{(O)}\) defined as follows
$$\begin{aligned}&\mathcal {G}^{(O)}: \left\langle \big \{h_i^{+},h^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \right\rangle \\&\quad = \left\langle \big \{(\partial _x+\partial _t^{1/2})e_{i+1}(\cdot ,\xi _{i}^{+}), (\partial _x-\partial _t^{1/2}) e_{i}(\cdot ,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\right\rangle \, . \end{aligned}$$
Thus, we can write that
$$\begin{aligned}&\widehat{\mathcal {G}}^{(O)}\big (\langle \{{\widehat{h}}_i^{+},{\widehat{h}}_{i+1}\}_{1 \leq i \leq m-1} \rangle \big )(\tau )\\&\quad = \left\langle \big \{ (\partial _x + \alpha (\tau )){\widehat{e}}_{i+1}(\tau ,\xi _{i}^{+}),(\partial _x - \alpha (\tau )){\widehat{e}}_{i}(\tau ,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\right\rangle \\&\quad = 2\alpha (\tau ) \left\langle \big \{A_{i+1}(\tau )e^{\alpha (\tau )\xi _{i}^{+}},-B_{i}(\tau )e^{-\alpha (\tau )\xi _{i+1}^{-}} \big \}_{1 \leq i \leq m-1} \right\rangle . \end{aligned}$$
Let us introduce
$$\begin{aligned} {\widehat{h}}_i^{+,(2)}(\tau )& = 2\alpha (\tau )A_{i+1}(\tau )e^{\alpha (\tau )\xi _{i}^{+}}, \\ {\widehat{h}}_{i+1}^{-,(2)}(\tau )& = -2\alpha (\tau )B_{i}(\tau )e^{-\alpha (\tau )\xi _{i+1}^{-}} \, , \end{aligned}$$
so that we get: \({\widehat{e}}^{(2)}_i(\tau ,x) = A^{(2)}_i(\tau )e^{\alpha (\tau )x} + B^{(2)}_i(\tau )e^{-\alpha (\tau )x}\). From the boundary conditions, we find, for \(i \in \{2,\ldots ,m-1\}\),
$$\begin{aligned} \left\{ \begin{array}{c} \displaystyle A^{(2)}_i(\tau ) = \frac{{\widehat{h}}^{+,(2)}_i(\tau )e^{-\alpha (\tau )\xi _{i}^+}}{2\alpha (\tau )} = A_{i+1}(\tau ),\\ \displaystyle B^{(2)}_i(\tau ) = -\frac{{\widehat{h}}^{-,(2)}_i(\tau )e^{\alpha (\tau )\xi _{i}^-}}{2\alpha (\tau )} = B_{i-1}(\tau ). \end{array} \right. \end{aligned}$$
In addition, we have
$$\begin{aligned}&\widehat{\mathcal {G}}^{(O),2}\big (\langle \big \{{\widehat{h}}_i^{+},{\widehat{h}}^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \rangle \big )(\tau )\\&\quad = \left\langle \big \{(\partial _x + \alpha (\tau )){\widehat{e}}^{(2)}_{i+1}(\tau ,\xi _{i}^{+}), (\partial _x - \alpha (\tau )){\widehat{e}}^{(2)}_{i}(\tau ,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\right\rangle \\&\quad = 2\alpha (\tau ) \left\langle \big \{ A^{(2)}_{i+1}(\tau )e^{\alpha (\tau )\xi _{i}^{+}},-B^{(2)}_{i}(\tau )e^{-\alpha (\tau )\xi _{i+1}^{-}} \big \}_{1 \leq i \leq m-1}\right\rangle \\&\quad = \left\langle \big \{ {\widehat{h}}_{i+2}^{+}(\tau )e^{-\alpha (\tau )(\xi _{i+2}^+-\xi _i^{+})},{\widehat{h}}_{i-1}^{-}(\tau )e^{-\alpha (\tau )(\xi _{i+1}^{-}-\xi _{i-1}^{-})} \big \}_{1 \leq i \leq m-1}\right\rangle \\&\quad = \left\langle \big \{ {\widehat{h}}_{i+2}^{+}(\tau )e^{-2\alpha (\tau )(L+\varepsilon )},{\widehat{h}}_{i-1}^{-}(\tau )e^{-2\alpha (\tau )(L+\varepsilon )} \big \}_{1 \leq i \leq m-1}\right\rangle . \end{aligned}$$
We can finally state the following result.

Proposition 2

The convergence rate\(C_{\varepsilon }^{(\text {O})}\)of the OSWRmethod with subdomains of lengthLis\(O\big (e^{-2\alpha (\tau )L}\big )\).

Remark 1

By using a simple unitary transformation, it is trivial to extend the above results to the case of a linear equation with time-dependent potential V(t) in \(L_{\text {loc}}^1(\mathbb {R})\). Indeed, from \(\big (\mathtt{i}\partial _t +\partial _x^{2} + V(t)\big )\phi (t,x) = 0\), it is sufficient to define the new unknown \({\widetilde{\phi }}(t,x) = e^{-\mathtt{i}\int _{0}^tV(s)}\phi (t,x)\) by gauge change, \({\widetilde{\phi }}\) satisfying then the potential-free Schrödinger equation. This idea was previously used in several papers (see e.g. [2, 4]).

The Space Variable Potential Case

We now assume that the potential is space-dependent. Then, the argument used in Remark 1 is no longer valid. As it whas studied in [9], we can no more get a simple expression of the exact solution on each subdomain. We then have to use approximations. Let us set : \(P(t,x,D) = \partial _t - \partial ^2_{x} -V(x)\), where V is smooth, bounded with bounded derivative.

CSWR Algorithm

Based on the same notations as above, we directly consider the error equation for the CSWR algorithm:
$$\begin{aligned} \left\{ \begin{array}{lll} \big (\mathtt{i}\tau - \partial _x^2 + V(x)\big ){\widehat{e}}_i(\tau ,x) = 0, \qquad (\tau ,x) \in \mathbb {R}\times \varOmega _i, \\ {\widehat{e}}_i(\tau ,\xi ^{\pm }) = {\widehat{h}}_i^{\pm }(\tau ), \qquad \tau \in \mathbb {R}. \end{array} \right. \end{aligned}$$
(4)
We also assume that V is positive, and that \(\Vert V\Vert _{\infty }\) exists. From Nirenberg’s factorization theorem [9, 26], we have
$$\begin{aligned} P(t, x,\partial _t, \partial _x) = (\partial _x+i\varLambda ^-)(\partial _x+i\varLambda ^+) + {\mathcal {R}} , \end{aligned}$$
(5)
where \(\mathcal {R} \in \text{ OPS }^{-\infty }\) is a smoothing pseudodifferential operator. The operators \(\varLambda ^{\pm }\) are pseudodifferential operators of order 1 / 2 (in time) and order zero in x. Furthermore, their total symbols \(\lambda ^{\pm }:=\sigma (\varLambda ^{\pm })\) can be expanded in the symbol class \(S^{1/2}_S\) as
$$\begin{aligned} \lambda ^{\pm } \sim \sum _{j=0}^{+\infty } \lambda _{1/2-j/2}^{\pm }, \end{aligned}$$
(6)
where \(\lambda _{1/2-j/2}^{\pm }\) are symbols corresponding to operators of order \(1/2-j/2\), see [9]. We denote by \({\mathfrak {e}}_i\) an approximate solution to (4) in \(\varOmega _i\) of the following form (neglecting the scattering effects)
$$\begin{aligned} \widehat{{\mathfrak {e}}}_i(\tau ,x)& = {\mathfrak {A}}_i(\tau )\exp \big (-\mathtt{i}\int ^x_{\xi _i^+}\lambda ^{+}(y,\tau )dy\big ) \\&\quad + {\mathfrak {B}}_i(\tau )\exp \big (-\mathtt{i}\int ^x_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big ). \end{aligned}$$
By construction, \(\lambda ^-=-\lambda ^+\) if we select \(\lambda _{1/2}^{+} = -\lambda _{1/2}^{-}\) (see [9]). Now, by using the boundary conditions, we find that, for \(i \in \{2,\ldots ,m-1\}\),
$$\begin{aligned} \left\{ \begin{array}{c} \displaystyle {\mathfrak {A}}_i(\tau ) = \frac{{\widehat{h}}_i^{+}(\tau ) - {\widehat{h}}_i^{-}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )} \, ,\\ \displaystyle {\mathfrak {B}}_i(\tau ) = \frac{{\widehat{h}}_i^{-}(\tau ) - {\widehat{h}}_i^{+}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )} \, . \end{array} \right. \end{aligned}$$
Let us introduce
$$\begin{aligned}&{\mathcal {G}}^{(C)}: \left\langle \big \{h_i^{+},h_{i+1}\big \}_{1 \leq i \leq m-1} \right\rangle \\&\quad = \left\langle \big \{{\mathfrak {e}}_i(\cdot ,\xi _{i+1}^-),{\mathfrak {e}}_{i+1}(\cdot ,\xi _{i}^{+}) \big \}_{1 \leq i \leq m-1}\right\rangle . \end{aligned}$$
Thus we have
$$\begin{aligned}&{\widehat{\mathcal {G}}}^{(C)}: \left\langle \{{\widehat{h}}_i^{+},{\widehat{h}}_{i+1}\}_{1 \leq i \leq m-1} \right\rangle \\&\quad = \left\langle \big \{\widehat{{\mathfrak {e}}}_i(\cdot ,\xi _{i+1}^-),\widehat{{\mathfrak {e}}}_{i+1}(\cdot ,\xi _{i}^{+})\big \}_{1 \leq i \leq m-1} \right\rangle \, . \end{aligned}$$
Next, some computations yield
$$\begin{aligned}&\widehat{{\mathfrak {e}}}_i(\tau ,x)\\&\quad = \frac{\big ({\widehat{h}}_i^{+}(\tau )- {\widehat{h}}_i^{-}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )\big )\exp \big (\mathtt{i}\int ^{x}_{\xi _i^+}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )} \\&\qquad + \frac{\big ({\widehat{h}}_i^{-}(\tau ) - {\widehat{h}}_i^{+}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int ^x_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )} \end{aligned}$$
and
$$\begin{aligned}&\widehat{{\mathfrak {e}}}_{i+1}(\tau ,\xi _i^{+})\\&\quad = \frac{\big ({\widehat{h}}_{i+1}^{+}(\tau )- {\widehat{h}}_{i+1}^{-}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _{i+1}^+}_{\xi _{i+1}^-}\lambda ^{-}(y,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int _{\xi _i^+}^{\xi _{i+1}^+}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _{i+1}^+}_{\xi _{i+1}^-}\lambda ^{-}(y,\tau )dy\big )} \\&\qquad + \frac{\big ({\widehat{h}}_{i+1}^{-}(\tau ) - {\widehat{h}}_{i+1}^{+}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _{i+1}^+}_{\xi _{i+1}^-}\lambda ^{-}(y,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int ^{\xi ^+_i}_{\xi _{i+1}^-}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _{i+1}^+}_{\xi _{i+1}^-}\lambda ^{-}(y,\tau )dy\big )} \\&\quad = \frac{\big ({\widehat{h}}_{i+1}^{+}(\tau )- {\widehat{h}}_{i+1}^{-}(\tau )\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int _{0}^{L}\lambda ^{-}(y+\xi _i^+,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^{-},\tau )dy\big )} \\&\qquad + \frac{\big ({\widehat{h}}_{i+1}^{-}(\tau ) - {\widehat{h}}_{i+1}^{+}(\tau )\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int ^{\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )}. \end{aligned}$$
Then, we get
$$\begin{aligned}&\widehat{{\mathfrak {e}}}_i(\tau ,\xi _{i+1}^{-}) \\&\quad = \frac{\big ({\widehat{h}}_i^{+}(\tau )- {\widehat{h}}_i^{-}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )\big )\exp \big (\mathtt{i}\int ^{\xi _{i+1}^-}_{\xi _i^+}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )} \\&\qquad + \frac{\big ({\widehat{h}}_i^{-}(\tau ) - {\widehat{h}}_i^{+}(\tau )\exp \big (-\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int ^{\xi _{i+1}^-}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{\xi _i^+}_{\xi _i^-}\lambda ^{-}(y,\tau )dy\big )} \\&\quad = \frac{\big ({\widehat{h}}_i^{+}(\tau )- {\widehat{h}}_i^{-}(\tau )\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int ^{\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i}^-,\tau )dy\big )} \\&\qquad + \frac{\big ({\widehat{h}}_i^{-}(\tau ) - {\widehat{h}}_i^{+}(\tau )\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )\big )\exp \big (-\mathtt{i}\int ^{L}_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )}. \end{aligned}$$
Let us set \({\widehat{h}}_i^{+,(2)}(\tau ) := \widehat{{\mathfrak {e}}}_i(\tau ,\xi _{i+1}^-)\) and \({\widehat{h}}_{i+1}^{-,(2)}(\tau ) := \widehat{{\mathfrak {e}}}_{i+1}(\tau ,\xi _{i}^+)\). Then we are led to
$$\begin{aligned}&\widehat{{\mathfrak {e}}}^{(2)}_{i+1}(\tau ,\xi _i^{+}) \\&\quad = \frac{\widehat{{\mathfrak {e}}}_{i+1}(\tau ,\xi _{i+2}^-)- \widehat{{\mathfrak {e}}}_{i+1}(\tau ,\xi _{i}^+)\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^{-},\tau )dy\big )}\\&\qquad \times \exp \big (-\mathtt{i}\int _{0}^{L}\lambda ^{-}(y+\xi _i^+,\tau )dy\big ) \\&\qquad + \frac{\widehat{{\mathfrak {e}}}_{i+1}(\tau ,\xi _{i}^+) - \widehat{{\mathfrak {e}}}_{i+1}(\tau ,\xi _{i+2}^-)\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big )} \\&\qquad \times \exp \big (-\mathtt{i}\int ^{\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big ) \\&\quad = {\widehat{h}}^-_{i+1}(\tau )\exp \big (-2\mathtt{i}\varepsilon \lambda ^{-}(\tau ,\xi _{i+1}^-)\big ) + \text {R}_1\big (\tau ,\varepsilon ,L,\max _{1 \leq i \leq m}\Vert {\widehat{h}}^{\pm }_i\Vert \big ) \end{aligned}$$
and
$$\begin{aligned}&\displaystyle \widehat{{\mathfrak {e}}}^{(2)}_i(\tau ,\xi _{i+1}^{-}) \\&\quad = \frac{\widehat{{\mathfrak {e}}}_i(\tau ,\xi _{i+1}^-)- \widehat{{\mathfrak {e}}}_{i}(\tau ,\xi _{i-1}^+)\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _{i}^-,\tau )dy\big )}\\&\qquad \times \exp \big (-\mathtt{i}\int ^{\varepsilon }_{0}\lambda ^{-}(y+\xi _{i+1}^-,\tau )dy\big ) \\&\qquad + \frac{\widehat{{\mathfrak {e}}}_{i}(\tau ,\xi _{i-1}^+) - \widehat{{\mathfrak {e}}}_i(\tau ,\xi _{i+1}^-)\exp \big (-\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )}{1-\exp \big (-2\mathtt{i}\int ^{L+\varepsilon }_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )} \\&\qquad \times \exp \big (-\mathtt{i}\int ^{L}_{0}\lambda ^{-}(y+\xi _i^-,\tau )dy\big )\\&\quad = {\widehat{h}}^+_i(\tau )\exp \big (-2\mathtt{i}\varepsilon \lambda ^{-}(\tau ,\xi _{i+1}^-)\big ) + \text {R}_2\big (\tau ,\varepsilon ,L,\max _{1 \leq i \leq m}\Vert {\widehat{h}}_i\Vert \big ), \end{aligned}$$
where \(\text {R}_{\ell } = O\big (\max _{1 \leq i \leq m}\Vert {\widehat{h}}_i\Vert e^{-\alpha (\tau )L}\big )\), \(\ell =1,2\). We then have
$$\begin{aligned}&\widehat{\mathcal {G}}_i^{(C),2}\big (\langle \{{\widehat{h}}^+_i(\tau ),{\widehat{h}}_{i+1}^-(\tau )\}_{i}\rangle \big ) \\&\quad = C_i(\varepsilon ,\tau ,V)\left\langle \big \{{\widehat{h}}_i^+(\tau ), {\widehat{h}}_{i+1}^-(\tau ) \big \}_{i}\right\rangle + O\big (\max _{1 \leq i \leq m}\Vert {\widehat{h}}_i\Vert e^{-\alpha (\tau )L}\big ). \end{aligned}$$
We yet refer [9], where for large \(\tau\), we can rigorously expand the symbol \(\lambda ^{\pm }\). We do not proceed to these laborious expansions in this paper. In the first approximation [9], the local convergence rate between \(\varOmega _{i},\) and \(\varOmega _{i+1}\) is given by
$$\begin{aligned} C_i(\varepsilon ,\tau ) \approx \exp \big (-\varepsilon \big (\sqrt{2|\tau |}+V(\xi _{i+1}^-))\big ). \end{aligned}$$
In general, the above procedure does not allow to directly construct a contraction factor, i.e. subdomain-independent coefficients \(C_i\). This is a consequence to the fact that the convergence rate is dependent on the values of the potential at the subdomain interfaces. However we argue that the contraction factor (related to the convergence rate of the CSWR method) \(C_{\varepsilon }^{(C)}\) is such that
$$\begin{aligned}&\sup \nolimits _{\tau }\exp \big (-\varepsilon (\sqrt{2|\tau |}+\max _{i \in \{1,\ldots ,m-1\} }V(\xi _{i+1}^-))\big ) \lesssim C_{\varepsilon }^{(C)} \\&\quad \lesssim \sup \nolimits _{\tau }\exp \big (-\varepsilon (\sqrt{2|\tau |} +\min _{i \in \{1,\ldots ,m-1\} }V(\xi _{i+1}^-))\big )\\&\quad \leq \sup \nolimits _{\tau }\exp \big (-\varepsilon \sqrt{2|\tau |}\big ) . \end{aligned}$$
The following Lemma actually justifies that the convergence rate obtained in the case of m subdomains is close to the one for two subdomains and numerically seen in [9] and in Sect. 4.

Lemma 1

Let us assume that\(f \, : \, {{\varvec{x}}} \in \mathbb {R}^N \mapsto f({{\varvec{x}}})\in \mathbb {R}^N\), is such that\(f({{\varvec{x}}}) = \gamma {{\varvec{x}}} + {\varvec{\varepsilon }}\), with\(\gamma <1\)and\(\Vert {\varvec{\varepsilon }}\Vert =o(\gamma )\). We define the sequence\({{\varvec{x}}}_{n+1} = f({{\varvec{x}}}_n) + {\varvec{\varepsilon }}\), with\({{\varvec{x}}}_0 \in \mathbb {R}^N\)given. Then, we have:\(\Vert {{\varvec{x}}}_{n}\Vert \leq \gamma ^{n}\Vert {{\varvec{x}}}_0\Vert + o(\gamma )\).

For \(\Vert V\Vert _{\infty }\) small enough, we directly get
$$\begin{aligned} C^{(C)}_{\varepsilon }\approx & {} \sup _{\tau }\exp \big (-2\mathtt{i}\varepsilon \alpha (\tau )\big ). \end{aligned}$$
For the sake of simplicity, we do not detail more the computations. We conclude that, as in the potential-free case the contraction factor for m sufficiently large subdomains (\(L \gg 1\)) is close to the one for the two-subdomains case.

Proposition 3

The convergence rate\(C_{\varepsilon }^{\text {(C)}}\)of the CSWR method withmsubdomains of lengthLis the same as for two-subdomains(see [9]) up to a term of the order of\(O\big (e^{-L\sqrt{2|\tau |}}\big )\).

Although, the convergence rate \(C_{\varepsilon }^{(\text {C})}\) is determined, the overall convergence is also proportional to the number of subdomains. In particular, the transmission from one subdomain to the next one naturally slows down the overall convergence of the SWR method, but without changing the slope of the residual history in logscale [9].

OSWR Algorithm

We now consider the OSWR algorithm. More specifically, from time \(t_n\) to \(t_{n+1^-}\), the OSWR algorithm with potential reads as follows:
$$\begin{aligned} \left\{ \begin{array}{l} P(\partial _t,\partial _x)\phi _i^{(k)} = 0, \quad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _i,\\ \phi _i^{(k)}(0,\cdot ) = \varphi _0, \quad x \in \varOmega _i, \\ \big (\partial _x+\mathtt{i}\varLambda ^+(\tau ,x)\big )\phi ^{(k)}_i(t,\xi _i^{+}) = \big (\partial _x+\mathtt{i}\varLambda ^+(\tau ,x)\big )\phi ^{(k-1)}_{i+1}(t,\xi _{i}^{+}), \quad t \in (t_n,t_{n+1}^-), \\ \big (\partial _x+\mathtt{i}\varLambda ^-(\tau ,x)\big )\phi ^{(k)}_i(t,\xi _i^{-}) = \big (\partial _x+\mathtt{i}\varLambda ^-(\tau ,x)\big )\phi ^{(k-1)}_{i-1}(t,\xi _{i}^{-}), \quad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
(7)
The operators \(\varLambda ^{\pm }\) are coming from (5) and the corresponding symbols are denoted by \(\lambda ^{\pm }\). The latter can be constructed as an asymptotic series of the form (6). In practice, the series \(\sum _{j=0}^{+\infty } \lambda _{1/2-j/2}^{\pm }\) is truncated and, for \(p \in \mathbb {N}\), we can define \(\lambda _p^{\pm } :=\sum _{j=0}^{p} \lambda _{1/2-j/2}^{\pm }\) and the corresponding operators \(\varLambda _p^{\pm }\). In [9], the SWR convergence rate is established for the transmission operator \(\partial _t \pm \mathtt{i} \varLambda ^{\pm }_p\). In this paper, we will only consider \(\varLambda ^{\pm }\) but the results could be extended to \(\varLambda ^{\pm }_p\) in a similar way. The error equation in Fourier (resp. real) space in time (resp. space) is
$$\begin{aligned} \left\{ \begin{array}{lll} \big (\mathtt{i}\tau -V(x) - \partial _x^2\big ){\widehat{e}}_i(\tau ,x) & = 0, \qquad (\tau ,x) \in \mathbb {R}\times \varOmega _i, \\ \quad \big (\partial _x + \mathtt{i}\varLambda ^{\pm }(\tau ,x)\big ){\widehat{e}}_i(\tau ,\xi ^{\pm }) & = {\widehat{h}}_i^{\pm }(\tau ), \qquad \tau \in \mathbb {R}. \end{array} \right. \end{aligned}$$
The convergence analysis is identical to the one that we presented in Sect. 2.2. Basically, we define
$$\begin{aligned}&\mathcal {G}^{(O)}: \left\langle \big \{h_i^{+},h^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \right\rangle \\&\quad = \left\langle \big \{\big (\partial _x+\mathtt{i}\varLambda ^+(\cdot ,x)\big )e_{i+1}(\cdot ,\xi _{i}^{+}),\right. \\&\left. \qquad \big (\partial _x+\mathtt{i}\varLambda ^-(\cdot ,x)\big ) e_{i}(t,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\right\rangle . \end{aligned}$$
Thus, we can define
$$\begin{aligned}&\widehat{\mathcal {G}}^{(O)}: \langle \big \{{\widehat{h}}_i^{+},{\widehat{h}}^-_{i+1}\big \}_{1 \leq i \leq m-1} \rangle \\&\quad = \langle \big \{ \big (\partial _x + \mathtt{i}\lambda ^+(\cdot ,x)\big ){\widehat{e}}_{i+1}(\cdot ,\xi _{i}^{+}),\\&\qquad \big (\partial _x+\mathtt{i}\lambda ^-(\cdot ,x)\big ){\widehat{e}}_{i}(\cdot ,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\rangle . \end{aligned}$$
We define approximate solutions to (7) on each \(\varOmega _i\), neglecting again the scattering effets
$$\begin{aligned} \widehat{{\mathfrak {e}}}_i(\tau ,x)& = {\mathfrak {A}}_i(\tau )\exp \big (-\mathtt{i}\int ^x_{0}\lambda ^{+}(y,\tau )dy\big ) \\&\quad + {\mathfrak {B}}_i(\tau )\exp \big (\mathtt{i}\int ^x_{0}\lambda ^{-}(y,\tau )dy\big ). \end{aligned}$$
Then by construction \(\lambda ^+=-\lambda ^-\), we get
$$\begin{aligned} {\mathfrak {A}}_i(\tau ) = \frac{{\widehat{h}}_i^+(\tau )e^{\int _0^{\xi _i^+}\lambda ^+(\tau ,y)dy}}{2\mathtt{i}\lambda ^+(\tau ,x)}, \qquad {\mathfrak {B}}_i(\tau ) = -\frac{{\widehat{h}}_i^-(\tau )e^{\int _0^{\xi _i^-}\lambda ^-(\tau ,y)dy}}{2\mathtt{i}\lambda ^+(\tau ,x)} \, , \end{aligned}$$
leading to
$$\begin{aligned}&\widehat{\mathcal {G}}^{(O),2}\big (\langle \big \{{\widehat{h}}_i^{+},{\widehat{h}}^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \rangle \big )(\tau )\\&\quad = \left\langle \big \{(\partial _x + \mathtt{i}\lambda ^{+}(\tau ,x)){\widehat{e}}^{(2)}_{i+1}(\tau ,\xi _{i}^{+}), (\partial _x - \mathtt{i}\lambda ^{+}(\tau ,x)){\widehat{e}}^{(2)}_{i}(\tau ,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\right\rangle \\&\quad = \left\langle \big \{ {\widehat{h}}_{i+2}^{+}(\tau )e^{-\mathtt{i}\int _{\xi _i^+}^{\xi _{i+2}^+}\lambda ^{+}(\tau ,y)},{\widehat{h}}_{i-1}^{-}(\tau )e^{-\mathtt{i}\int _{\xi _{i-1}^-}^{\xi _{i+1}^-}\lambda ^{+}(\tau ,y)dy} \big \}_{1 \leq i \leq m-1}\right\rangle \\&\quad = \left\langle \big \{ {\widehat{h}}_{i+2}^{+}(\tau )e^{-2\mathtt{i}\int _0^{L+\varepsilon }\lambda ^{+}(\tau ,y+\xi _i^+)} ,{\widehat{h}}_{i-1}^{-}(\tau )e^{-2\mathtt{i}\int _0^{L+\varepsilon }\lambda ^{+}(\tau ,y+\xi _{i-1}^-)dy} \big \}_{1 \leq i \leq m-1}\right\rangle \, . \end{aligned}$$
Similarly to the CSWR method, the OSWR DDM asymptotically has a convergence rate mainly independent, up to a multiplicative constant, of the number of subdomains of length L. Details of the analysis for the convergence over 2 subdomains for the OSWR and quasi-OSWR can be found in Sect. 2.3 of [9].

Scalability

We notice that using a large number of subdomains does not allow for an acceleration of the convergence rate, which is the same as for the two subdomains case up to a multiplication coefficient. We however benefit from i) an embarrassingly parallel algorithm and ii) local computations on each subdomain.

Extension to Time-Dependent Problems

The principle for analyzing the rate of convergence in the time-dependent equation is closely related to the stationary case (see also [10]). Basically, we have to replace t (resp. \(\tau\)) by \(\mathtt{i}t\) (resp. \(\mathtt{i}\tau\)) in the equations. Let \(Q(\partial _t,\partial _x) = \mathtt{i}\partial _t + \partial ^2_{x}\) be the Schrödinger operator in real time. Let us consider the IBVP on \(\varOmega =(-a,a)\)
$$\begin{aligned} \left\{ \begin{array}{lll} \big (\mathtt{i}\partial _t + \partial ^2_{x}\big ) \phi (t,x) & = 0, \qquad (t,x) \in (0,T)\times \varOmega , \\ \phi (0,\cdot ) & = \varphi _0, \qquad x \in \varOmega , \\ \phi (t,-a) & = 0, \qquad t \in (0,T),\\ \phi (t,a) & = 0, \qquad t \in (0,T), \end{array} \right. \end{aligned}$$
where \(T>0\) and \(\varphi _0 \in L^2(\mathbb {R})\) are given. The convergence rate is defined as the slope of the logarithm residual history according to the Schwarz iteration number, that is \(\{(k,\log (\mathcal {E}^{(k)})) \, : \, k \in \mathbb {N}\}\), where now
$$\begin{aligned}\mathcal {E}^{(k)} &:= \sum _{i=1}^{m-1}\big \Vert \ \ \Vert \phi ^{\text {(k)}}_{i\big |(\xi _{i+1}^-,\xi _i^+)}\nonumber \\&\quad -\phi ^{\text {(k)}}_{i+1\big |(\xi _{i+1}^-,\xi _i^+)}\Vert _{\infty }\big \Vert _{L^{2}(0,T)}\leq \delta ^{\text {Sc}}, \end{aligned}$$
(8)
\(\delta ^{\text {Sc}}\) being a small parameter.
First, the CSWR algorithm reads as follows, for \(k \geq 1\) and for \(i \in \{2,\ldots ,m-1\}\),
$$\begin{aligned} \left\{ \begin{array}{lll} Q(\partial _t,\partial _x)\phi _i^{(k)} & = 0, \qquad (t,x) \in (0,T)\times \varOmega _i,\\ \phi _i^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _i, \\ \phi ^{(k)}_i(t,\xi _i^{+}) & = \phi ^{(k-1)}_{i+1}(t,\xi _{i}^{+}), \qquad t \in (0,T), \\ \phi ^{(k)}_i(t,\xi _i^{-}) & = \phi ^{(k-1)}_{i-1}(t,\xi _{i}^{-}), \qquad t \in (0,T). \end{array} \right. \end{aligned}$$
In \(\varOmega _1\), we get
$$\begin{aligned} \left\{ \begin{array}{lll} Q(\partial _t,\partial _x)\phi _1^{(k)} & = 0, \qquad (t,x) \in (0,T)\times \varOmega _1,\\ \phi _1^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _1, \\ \phi ^{(k)}_1(t,-a) & = 0, \qquad t \in (0,T), \\ \phi ^{(k)}_1(t,\xi _1^{+}) & = \phi ^{(k-1)}_{2}(t,\xi _{2}^{+}), \qquad t \in (0,T), \end{array} \right. \end{aligned}$$
and in \(\varOmega _m\)
$$\begin{aligned} \left\{ \begin{array}{lll} Q(\partial _t,\partial _x)\phi _m^{(k)} & = 0, \qquad (t,x) \in (0,T)\times \varOmega _m,\\ \phi _m^{(k)}(0,\cdot ) & = \varphi _0, \qquad x \in \varOmega _m, \\ \phi ^{(k)}_i(t,\xi _m^{-}) & = \phi ^{(k-1)}_{m-1}(t,\xi _{m}^{-}), \qquad t \in (0,T), \\ \phi ^{(k)}_m(t,+a) & = 0, \qquad t \in (0,T). \end{array} \right. \end{aligned}$$
In order to analyze the convergence of the SWR methods, it is sufficient to replace \(\tau\) by \(\mathtt{i}\tau\) in all the equations from Sect. 2. In particular, for m subdomains, we can write that
$$\begin{aligned} \widehat{\mathcal {G}}^{(C),2}\big (\langle \{{\widehat{h}}^+_i,{\widehat{h}}_{i+1}^-\}_{i}\rangle \big )(\tau )& = e^{-2\beta (\tau )\varepsilon }\langle \big \{{\widehat{h}}_i^+(\tau ), {\widehat{h}}_{i+1}^-(\tau ) \big \}_i \rangle \\&\quad + O\big (e^{-\alpha (\tau )L}\big ), \end{aligned}$$
where \(\beta (\tau ) := \sqrt{\tau }\).
A similar study is possible with OSWR algorithm for m subdomains. First, let us define the transparent operator \(\partial _x\pm e^{-\mathtt{i}\pi /4}\partial _t^{1/2}\) at the interfaces. The OSWR algorithm reads as follows: for \(k \geq 1\) and for \(i \in \{2,\ldots ,m-1\}\)
$$\begin{aligned} \left\{ \begin{array}{l} Q(\partial _t,\partial _x)\phi _i^{(k)} = 0, \qquad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _i,\\ \phi _i^{(k)}(0,\cdot ) = \varphi _0, \qquad x \in \varOmega _i, \\ (\partial _x+e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k)}_i(t,\xi _i^{+}) = (\partial _x+e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k-1)}_{i+1}(t,\xi _{i}^{+}), \quad t \in (t_n,t_{n+1}^-), \\ (\partial _x-e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k)}_i(t,\xi _i^{-}) = (\partial _x-e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k-1)}_{i-1}(t,\xi _{i}^{-}), \quad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
In \(\varOmega _1\), we get
$$\begin{aligned} \left\{ \begin{array}{l} Q(\partial _t,\partial _x)\phi _1^{(k)} = 0, \quad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _1,\\ \phi _1^{(k)}(0,\cdot ) = \varphi _0, \quad x \in \varOmega _1, \\ \phi ^{(k)}_1(t,-a) = 0, \quad t \in (t_n,t_{n+1}^-), \\ (\partial _x+e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k)}_1(t,\xi _1^{+}) = (\partial _x+e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k-1)}_{2}(t,\xi _{1}^{+}), \quad t \in (t_n,t_{n+1}^-), \end{array} \right. \end{aligned}$$
and in \(\varOmega _m\)
$$\begin{aligned} \left\{ \begin{array}{l} Q(\partial _t,\partial _x)\phi _m^{(k)} = 0, \quad (t,x) \in (t_n,t_{n+1}^-)\times \varOmega _m,\\ \phi _m^{(k)}(0,\cdot ) = \varphi _0, \quad x \in \varOmega _m, \\ (\partial _x-e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k)}_i(t,\xi _m^{-}) = (\partial _x-e^{-\mathtt{i}\pi /4}\partial _t^{1/2})\phi ^{(k-1)}_{m-1}(t,\xi _{m}^{-}), \quad t \in (t_n,t_{n+1}^-), \\ \phi ^{(k)}_m(t,+a) = 0, \quad t \in (t_n,t_{n+1}^-). \end{array} \right. \end{aligned}$$
We then consider
$$\begin{aligned} \left\{ \begin{array}{l} \big (\mathtt{i}\tau - \partial _x^2\big ){\widehat{e}}_i(\tau ,x) = 0, \qquad (\tau ,x) \in \mathbb {R}\times \varOmega _i, \\ (\partial _x \pm \beta (\tau )){\widehat{e}}_i(\tau ,\xi ^{\pm }) = {\widehat{h}}_i^{\pm }(\tau ), \qquad \tau \in \mathbb {R}, \end{array} \right. \end{aligned}$$
where we have set \(\beta (\tau ) := e^{\mathtt{i}\pi /4}\sqrt{\tau }\). We get:
$$\begin{aligned} {\widehat{e}}_i(\tau ,x) = A_i(\tau )e^{\beta (\tau )x} + B_i(\tau )e^{-\beta (\tau )x}. \end{aligned}$$
From the boundary conditions, we find, for \(i \in \{2,\ldots ,m-1\},\)
$$\begin{aligned} A_i(\tau ) = \frac{{\widehat{h}}_i^+(\tau )e^{-\beta (\tau )\xi _{i}^+}}{2\beta (\tau )}, \qquad B_i(\tau ) = -\frac{{\widehat{h}}_i^-(\tau )e^{\beta (\tau )\xi _{i}^-}}{2\beta (\tau )}. \end{aligned}$$
We introduce a mapping \(\mathcal {G}^{(O)}\) defined as follows
$$\begin{aligned}&\mathcal {G}^{(O)}: \left\langle \big \{h_i^{+},h^{-}_{i+1}\big \}_{1 \leq i \leq m-1} \right\rangle \\&\quad = \left\langle \big \{(\partial _x+e^{-\mathtt{i}\pi /4}\partial _t^{1/2})e_{i+1}(\cdot ,\xi _{i}^{+}),\right. \\&\left. \qquad (\partial _x-e^{-\mathtt{i}\pi /4}\partial _t^{1/2}) e_{i}(\cdot ,\xi _{i+1}^-) \big \}_{1 \leq i \leq m-1}\right\rangle . \end{aligned}$$
We again find the same convergence rate as in [10] for two subdomains, but the overall error is still shifted in logscale, by a positive constant linearly dependent on \(\log (m)\).

Numerical Experiments

Eigenvalue Problem

We present a simple test case illustrating the theoretical results presented in Sect. 2. We take \(V(x)=5x^2/2\), and \(\varOmega =(-2,2)\) and solve the Schrödinger equation in imaginary-time by using a three-point finite difference scheme, with meshsize \(\varDelta x=1/64\), and for a time step \(\varDelta t=0.2\). Details about the solver implementation can be found in [9]. We apply the CSWR method on \(m=2\), 4, 8 subdomains and compare the convergence rate as a function of the Schwarz iterations. Initially, we take \(\phi _0(x)={\widetilde{\phi }}_0(x)/\Vert {\widetilde{\phi }}_0\Vert _2\), with \({\widetilde{\phi }}_0(x)=e^{-4x^2}\). The \(m-1\) overlapping zones have a length equal to \(\varepsilon =\varDelta x\), corresponding to 2 nodes. We observe on Fig. 2 that the asymptotic residual history is numerically independent of the number of subdomains.
Fig. 2

Comparison of the asymptotic convergence rates for 2, 4 and 8 subdomains, including a zoom in the asymptotic regime

We also report on Fig. 3 the converged solution \(\big \{(x,\phi ^{\text {cvg},(k^{(\text {cvg})})}(x)=\phi _g(x)), \, x\in (-2,2)\big \}\), the initial guess \(\big \{(x,\phi _0(x)), \, x\in (-2,2)\big \}\), as well as the NGF converged solution after 20 Schwarz iterations with 8 subdomains, \(\{(x,\phi ^{\text {cvg},(20)}(x)), \, x\in (-2,2)\}\). We however numerically observe that for 8 subdomains the asymptotic rates of convergence seems to be relatively different than 2 and 4. This can be explained by the fact that, for a large number of subdomains, the size of each of these subdomains is relatively small, which induces an inaccuracy in the theoretical slope of convergence. In addition, the overall convergence rate depends on the value of the potential at the subdomain interfaces. In the next example, the overall domain is larger, leading to larger subdomains, then better accuracy of the convergence rate.
Fig. 3

Initial guess, potential and converged ground state

Time-Dependent Equation

In this example, the potential is chosen as \(V(x)=-5\exp (-2x^2)\), with \(\varOmega =(-8,8)\) and final time \(T=10\). We solve the Schrödinger equation in real-time with the three-point finite difference scheme, for \(\varDelta x=1/32\) and \(\varDelta t=10^{-2}\). We refer to [10] for the details concerning the solver. The CSWR method is applied for \(m=2\), 4, 8 and 16 subdomains. We compare the convergence rate as a function of the Schwarz iterations. The initial data is defined by \(\phi _0(x)={\widetilde{\phi }}_0(x)/\Vert {\widetilde{\phi }}_0\Vert _2\), with \({\widetilde{\phi }}_0(x)=e^{-5(x+2)^2+\mathtt{i}k_0x}\), for \(k_0=5\). The \(m-1\) overlapping zones have a length equal to \(\varepsilon =\varDelta x\), corresponding to 2 nodes. We observe on Fig. 4 that, as proven above, the asymptotic residual history is essentially independent of the number of subdomains.
Fig. 4

Comparison of the asymptotic convergence rates for 2, 8, 16 and 32 subdomains, including a zoom in the asymptotic regime

We also report in Fig. 5 the amplitude of the converged solution \(\big \{(x,|\phi ^{N,(k^{(\text {cvg})})}(x)|, \, x\in (-8,8)\big \}\), the initial guess \(\big \{(x,|\phi _0(x)|), \, x\in (-8,8)\big \}\), as well as the TDSE converged solution after 10 Schwarz iterations with 16 subdomains, and finally \(\big \{(x,|\phi ^{N,(7)}(x)|), \, x\in (-8,8)\big \}\), where N is such that \(T_N=T=10\).
Fig. 5

Amplitudes of the initial guess, converged solution and solution after 7 Schwarz iterations on 16 subdomains

Conclusion

In this paper, we presented an asymptotic analysis of the convergence rate of the multi-domains CSWR and OSWR DDMs for the one-dimensional linear Schrödinger equation in imaginary- and real-time. Asymptotic estimates show that the already existing estimates stated for the two-domains configuration extend to \(m > 2\) domains. This is illustrated through some numerical examples.

Notes

Acknowledgements

X. Antoine was supported by the French National Research Agency project NABUCO, Grant ANR-17-CE40-0025. E. Lorin thanks NSERC for the financial support via the Discovery Grant program.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Institut Elie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, Inria Nancy-Grand Est, SPHINX TeamVandoeuvre-lès-Nancy CedexFrance
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  3. 3.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada

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