Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation

Abstract

This paper presents a coherent analysis of the finite difference method to nonlinear Schrödinger (NLS) equations in one spatial dimension. We use the discrete $H^1$ framework to establish well-posedness and error estimates in the $L^\infty$ norm. The nonlinearity f(u) of a NLS equation is assumed to satisfy only a growth condition. We apply our results to computation of blow-up solutions for a NLS equation with the nonlinearity $f(u)=-|u|^{2p}$, p being a positive real number. Particularly, we offer the numerical blow-up time $T(h,\tau )$, where h and $\tau$ are discretization parameters of space and time variables. We prove that $T(h,\tau )$ converges to the blow-up time $T_\infty$ of the solution of the original NLS equation. Several numerical examples are presented to confirm the validity of theoretical results. Furthermore, we infer from numerical investigation that the convergence of $T(h,\tau )$ is at a second order rate in $\tau$ if the Crank–Nicolson scheme is applied to time discretization.

Appendices

Appendix 1: Well-posedness of (13)

The following results are not new for specialists of nonlinear Schrödinger equations. For example, Proposition 7.1 is fundamentally described in [5, Theorem 3.5.1]. However, results for more regular solutions are not given explicitly in [5]. If considering the Cauchy problem, we can use the smoothing property of the Schrödinger semigroup and obtain a regular (global-in-time) solution in a certain sense (see [11]). However, in the case of a bounded domain, any smoothing properties are not available. Therefore, we assume sufficiently smooth data f, g and $u_0$ to obtain a smooth solution.

Let $I=(0,L)$ for $L>0$. Any function spaces considered in this appendix are still complex-valued. We introduce the following linear operators A and H of $L^2(I)\rightarrow L^2(I)$ by

$$\begin{aligned} \mathcal {D}(A)= & {} H^2(I)\cap H_0^1(I),\quad A v=-\frac{d^2}{dx^2}v\quad (v\in \mathcal {D}(A)),\\ \mathcal {D}(H)= & {} \mathcal {D}(A),\quad Hv=iA v\quad (v\in \mathcal {D}(H)). \end{aligned}$$

The following results are well known (see [16]). Operator A is a positive and self-adjoint operator in $L^2(I)$ and $-A$ generates the analytic semigroup (of class $C_0$) $e^{-tA}$ in $L^2(I)$. The operator $-H$ generates a $C_0$ semigroup $S(t)=e^{-itH}$ in $L^2(I)$ and $\Vert S(t)\Vert _{L^2(I)}=1$ for all $t>0$.

When $I=\mathbb {R}$, one can prove

$$\begin{aligned} \Vert S(t)\Vert _{H^1(I)}=1\quad (t>0) \end{aligned}$$

by the Fourier transform. However, it is not readily apparent that this equality remains valid if I is a bounded interval. Instead, we apply fractional powers $A^\frac{1}{2}$ of A. We know that

$$\begin{aligned} \mathcal {D}(A^{\frac{1}{2}})=H^1_0(I). \end{aligned}$$

Therefore, as a norm of $H_0^1(I)$, we can choose

$$\begin{aligned} |||v|||= \Vert A^{\frac{1}{2}}v\Vert _{L^2(I)}=\Vert \partial _x v\Vert _{L^2(I)}\, (v\in H_0^1(I)). \end{aligned}$$

By Poincaré’s inequality, we have

$$\begin{aligned} C^{-1}\Vert v\Vert _{H^1(I)}\le |||v|||\le C\Vert v\Vert _{H^1(I)}\quad (v\in H^1_0(I)). \end{aligned}$$

Then, we deduce the following results.

• $|||S(t)|||=1$ for all $t>0$.

• There exists a constant $C_I>0$ such that $\Vert v\Vert _{L^\infty (I)}\le C_I|||v|||$ for $v\in H_0^1(I)$.

We make the following condition on the nonlinearity f of $H^1_0(I)\rightarrow L^2(I)$.

Condition (f1) There exists a continuous, non-decreasing and positive function $\omega (\eta )$ of $\eta >0$ such that

$$\begin{aligned} \Vert A^{\frac{1}{2}}f(u)-A^{\frac{1}{2}}f(v)\Vert _{L^2(I)} \le \omega (M)\Vert A^{\frac{1}{2}}u-A^{\frac{1}{2}}v\Vert _{L^2(I)} \end{aligned}$$

for any $u,v\in \mathcal {D}(A^{\frac{1}{2}})$ with $\Vert A^{\frac{1}{2}}u\Vert _{L^2(I)},\Vert A^{\frac{1}{2}}v\Vert _{L^2(I)}\le M$ and $M>0$. This inequality is written equivalently as

$$\begin{aligned} \Vert \partial _x f(u)-\partial _x f(v)\Vert _{L^2(I)} \le \omega (M)\Vert \partial _x u-\partial _x v\Vert _{L^2(I)} \end{aligned}$$

for any $u,v\in H_0^1(I)$ with $\Vert u\Vert _{H^1(I)},\Vert v\Vert _{H^1(I)}\le M$.

Proposition 7.1

Assume that Condition (f1) is satisfied and that $g\in C^0([0,T];H^1_0(I))$. Then, for any $u_0\in H_0^1(I)$, there exists $T>0$ and a unique

$$\begin{aligned} u\in C^0([0,T];H^1_0(I))\cap C^1((0,T);[H^{1}_0(I)]') \end{aligned}$$

that satisfies

$$\begin{aligned}&\int _I i(\partial _tu)v~dx+\int _I (\partial _xu)(\partial _xv)~dx=\int _I f(u)v~dx+\int _I gv~dx\quad (\forall v\in H^1_0(I),\\&\quad \text{ a.e. } t\in (0,T)) \end{aligned}$$

with $u(0,x)=u_0(x)$ for $x\in I$. Moreover, if we define the maximal existence time $T_\infty$ as $T_\infty =\sup T$, then $T_\infty <\infty$ implies $\displaystyle {\lim _{t\rightarrow T_\infty }|||u(t)|||=\infty }$.

We then make the following condition for $f:H^1_0(I)\rightarrow L^2(I)$ and $2\le m\in \mathbb {Z}$.

Condition (f m ) For $k=1,\ldots ,m$ and $u\in \mathcal {D}(A^{m/2})$, we have $f(u)\in \mathcal {D}(A^{k/2})$. Moreover, there exists a continuous, non-decreasing and positive function $\omega _k(\eta )$ of $\eta >0$ such that

$$\begin{aligned} \Vert A^{k/2}[f(u)-f(v)]\Vert _{L^2(I)} \le \omega _k(M)\Vert A^{k/2}(u-v)\Vert _{L^2(I)} \end{aligned}$$

for any $u,v\in \mathcal {D}(A^{k/2})$ with $\Vert A^{k/2}u\Vert _{L^2(I)},\Vert A^{k/2}v\Vert _{L^2(I)}\le M$.

Proposition 7.2

Let $2\le m\in \mathbb {Z}$. Assume that Condition (fm) is satisfied and that $g\in C^{[m/2]}([0,\infty );H^1_0(I))$. Then, for any $u_0\in \mathcal {D}(A^{m/2})$, there exists $T>0$ and a unique

$$\begin{aligned} u\in \bigcap _{k=0}^{[m/2]}C^k([0,T];\mathcal {D}(A^{m/2-k})) \end{aligned}$$

that satisfies

$$\begin{aligned} i\partial _tu+\partial _x^2u=f(u)+g(t,x) \quad (0<t<T,~x\in I) \end{aligned}$$

with $u(0,t)=u_0(x)$ for $x\in I$.

Those Propositions 7.1 and 7.2 are proved fundamentally using the same method used by Segal [18] (see also [5, proof of Theorem 3.3.1]).

Remark 7.3

For $m\ge 1$, $u_0\in \mathcal {D}(A^{m/2})$ implies that $u_0=\partial _xu_0=\cdots =\partial _x^mu_0=0$ at $x=0,L$.

Appendix 2: Proof of Proposition 2.3

Let $f:\mathbb {C}\rightarrow \mathbb {C}$. Suppose that $\phi ({\xi ,\eta })={\text {Re}}f(z)$ and $\psi ({\xi ,\eta })={\text {Im}}f(z)$, $z={\xi +i\eta }$, are both $C^1$ functions of $\mathbb {R}_{{\xi }}\times \mathbb {R}_{{\eta }}\rightarrow \mathbb {R}$ and that $f(0)=0$ holds. First, we introduce a useful expression. We write

$$\begin{aligned} Df(z)=(\phi _{{\xi }}({\xi ,\eta }),\phi _{{\eta }}({\xi ,\eta }))^{\mathrm {T}}+i(\psi _{{\xi }}({\xi ,\eta }),\psi _{{\eta }}({\xi ,\eta }))^{\mathrm {T}}\qquad (z={\xi +i\eta }). \end{aligned}$$

In addition, for $w={{a}+i{b}}$, define $Df(z)\cdot w$ as

$$\begin{aligned} Df(z)\cdot w=\phi _{{\xi }}({\xi ,\eta }){a}+\phi _{{\eta }}({\xi ,\eta }){b} + i[\psi _{{\xi }}({\xi ,\eta }){a}+\psi _{{\eta }}({\xi ,\eta }){b}]. \end{aligned}$$

Then, if setting

$$\begin{aligned} |Df(z)|=\sqrt{\phi _{{\xi }}({\xi ,\eta })^2+\phi _{{\eta }}({\xi ,\eta })^2+\psi _{{\xi }}({\xi ,\eta })^2+\psi _{{\eta }}({\xi ,\eta })^2}, \end{aligned}$$

we have

$$\begin{aligned} |Df(z)\cdot w|\le |Df(z)|\cdot |w|\quad (z,~w\in \mathbb {C}). \end{aligned}$$

Furthermore, for $R>0$, setting

$$\begin{aligned} \alpha (R)=\max _{|z|\le \sqrt{L}R} |Df(z)|, \end{aligned}$$

we can estimate as

$$\begin{aligned} |Df(z)-Df(w)|\le 2\alpha (R)\quad (|z|,|w|\le \sqrt{L}R). \end{aligned}$$

(91)

Let $\varvec{u}=(u_1,\ldots ,u_N)^{\mathrm {T}},\varvec{v}=(v_1,\ldots ,v_N)^{\mathrm {T}}\in \mathbb {C}^N$ and $u_0=u_{N+1}=v_0=v_{N+1}=0$. Then,

$$\begin{aligned} q_j&\equiv f(u_j)-f(v_j)-[f(u_{j-1})-f(v_{j-1})]\\&=\int _0^1 Df(p_j(s))\cdot [(u_j-v_j)-(u_{j-1}-v_{j-1})]~ds\\&\quad +\int _0^1 [Df(p_j(s))-Df(p_{j-1}(s))]\cdot (u_{j-1}-v_{j-1})~ds \end{aligned}$$

for $1\le j\le N$, where $p_j(s)=sv_j+(1-s)u_j$.

At this stage, we let $R=|||\varvec{u}|||_h\wedge |||\varvec{v}|||_h$. Then, in view of Proposition 2.1 (i), we have $|p_j(s)|,|p_{j-1}(s)|\le \sqrt{L}R$ and we can estimate as

$$\begin{aligned} |q_j|^2 \le 2\alpha (R)^2 [(u_j-v_j)-(u_{j-1}-v_{j-1})]^2+4\alpha (R)^2L|||\varvec{u}-\varvec{v}|||_h^2. \end{aligned}$$

Therefore,

$$\begin{aligned} |||\varvec{f}(\varvec{u})-\varvec{f}(\varvec{v})|||_h^2&=\sum _{j=1}^{N+1}\frac{|q_j|^2}{h^2}h\\&\le 2\alpha (R)^2\sum _{j=1}^{N+1}\frac{|(u_j-v_j)-(u_{j-1}-v_{j-1})|^2}{h^2}h\\&\quad +4\alpha (R)^2L^2|||\varvec{u}-\varvec{v}|||_h^2\\&= (2+4L^2) \alpha (R)^2|||\varvec{u}-\varvec{v}|||_h^2, \end{aligned}$$

which implies (21b) with $C_{2f}(R)=\sqrt{2+4L^2}\alpha (R)=c_0(R)$.

Finally, (21a) follows by setting $\varvec{v}=\varvec{0}$.

Appendix 3: Modified Newton method

To solve our finite difference scheme (14), at each time step $t_n$, we must solve a nonlinear equation of the form

$$\begin{aligned} \mathcal {F}\varvec{u}= H\varvec{u}-K\varvec{v}+i{\Delta t}\left[ (1-\theta )\varvec{f}(\varvec{v})+\theta \varvec{f}(\varvec{u})+\varvec{g}^{n+\theta }\right] =\varvec{0}, \end{aligned}$$

(92)

where $\varvec{u}=\varvec{u}^{n+1}$, $\varvec{v}=\varvec{u}^{n}$, ${\Delta t}=\Delta t_n$, $H=H_n$, and $K=K_n$.

If decomposing the Eq. (92) into the real and imaginary parts, then we can apply any iterative methods for solving the system of equations of real functions. The standard Newton method is a powerful method. Another method is proposed in [2, §5]. However, if the nonlinearity f(z) is differentiable in the complex sense, we can use the complex Newton method (for the system of equations of complex functions). Consequently, MATLAB and Scilab are available to compute (92) using complex variables. However, $f(z)=\alpha z|z|^m$ and $f(z)=\alpha |z|^m$, $\alpha \in \mathbb {C}$, $m\ge 2$, are not differentiable in the complex sense so that the complex Newton method is not available. Instead, we offer a new iterative method that is a version of modified Newton methods to solve (92) using complex variables.

That is, we consider the following iteration: For an initial guess $\varvec{u}_0\in \mathbb {C}^N$, we generate $\{\varvec{u}_k\}_{k\ge 1}$ by

$$\begin{aligned} \varvec{u}_{k+1}&=\mathcal {N}\varvec{u}_k\nonumber \\&\equiv \varvec{u}_k-H^{-1} \mathcal {F}\varvec{u}_k \nonumber \\&=-i{\Delta t}\theta H^{-1}\varvec{f}(\varvec{u}_k)+H^{-1} [K\varvec{v}-i{\Delta t}(1-\theta )\varvec{f}(\varvec{v})-i{\Delta t} \varvec{g}^{n+\theta }]. \end{aligned}$$

(93)

This iterative method actually converges with a sufficiently small ${\Delta t}$, as stated in Proposition 9.1. Set $B_{R}=\{z\in \mathbb {C}\mid |z|\le R\}$ and take $\tilde{g}>0$ satisfying $|||\varvec{g}^{n}|||_h\le \tilde{g}$ for $0\le t_n\le T$.

Proposition 9.1

Assume that Condition (f) is satisfied. Let $\varvec{v}\in \mathbb {C}^N$ and $R=|||\varvec{v}|||_h$. Then, if

$$\begin{aligned} {\Delta t}\le \min \left\{ \frac{R}{~3\theta C_{1f}(2R)~},\ \frac{R}{~3(1-\theta )C_{1f}(R)~},\ \frac{R}{~3\tilde{g}~},\ \frac{1}{~2\theta C_{2f}(2R)~}\right\} , \end{aligned}$$

(94)

then $\mathcal {N}$ is a contraction mapping from $B_{2R}$ to $B_{2R}$. Consequently, there exists a unique fixed point $\varvec{u}\in B_{2R}$.

The proof is a direct consequence of Condition (f).

Example 9.2

We ignore the contribution of $\tilde{g}$ to set $\Delta t$ appropriately because $\tilde{g}$ is not so large relative to R. Consider $f(z)=\alpha u|u|^m$, $\alpha \in \mathbb {C}$ and $m\ge 2$. Then, (94) is written equivalently as

$$\begin{aligned} \Delta t \le \min \left\{ \frac{1}{3\theta c_1 2^{m+1}},\ \frac{1}{3(1-\theta )c_1}\right\} \frac{1}{R^m}, \end{aligned}$$

(95)

where $c_1$ is the constant defined in Example 2.4. Therefore, in this case, to apply the iterative method (93), we must take

$$\begin{aligned} \Delta t_{k} =\tau \min \left\{ 1,\ \frac{1}{|||\varvec{u}^n|||_h^m}\right\} ,\quad 0<\tau \le \tau ', \end{aligned}$$

(96)

where

$$\begin{aligned} \tau '=\gamma \min \left\{ \frac{1}{3\theta c_1 2^{m+1}},\ \frac{1}{3(1-\theta )c_1}\right\} \end{aligned}$$

(97)

and $\gamma$ is a constant taken from $0<\gamma <1$. For the case $f(z)=\alpha |z|^m$, $c_1$ should be replaced by $c_2$ in Example 2.4.

Remark 9.3

In view of (96), we must choose $q=2p$ in (34) to solve (32). However, we have verified from numerical experimentation that a modified Newton method always converges by setting $q=p$.