Optimal Convergence and Long-Time conservation of Exponential Integration for Schrödinger Equations in a Normal or Highly Oscillatory Regime

Abstract

In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schrödinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence and long time near conservations of density, momentum and actions is formulated and analysed. To this end, we propose continuous-stage exponential integrators and show that the integrators can exactly preserve the energy of Hamiltonian systems. Three practical energy-preserving integrators are presented. We establish that these integrators exhibit optimal convergence and have near conservations of density, momentum and actions over long times. A numerical experiment is carried out to support all the theoretical results presented in this paper. Some applications of the integrators to other kinds of ordinary/partial differential equations are also discussed.

Appendix

Proof of Lemma 1

Firstly, according to the scheme (2.4), the Duhamel principle (2.2) and the fact that

$$\begin{aligned} \left\| C_{\tau }({\mathcal {W}})w(\kappa _n)-e^{\text{ i }\tau \delta \kappa \triangle }w(\kappa _n)\right\| _{H^{\alpha -2}}\lesssim \delta \kappa , \end{aligned}$$

it is clearly that $\Vert \delta ^{n+\tau }\Vert _{H^{\alpha -2}}\lesssim \delta \kappa .$ Then it follows from the Duhamel principle (2.2) that

$$\begin{aligned} \begin{aligned} w(\kappa _n+\tau \delta \kappa )&=e^{\text{ i }\tau \delta \kappa \triangle }w(\kappa _n)+ \varepsilon \tau \delta \kappa \varphi _{1}(\tau {\mathcal {W}})f(w(\kappa _n))\\&\quad + \varepsilon \tau ^2 \delta \kappa ^2 \int _{0}^1 \int _{0}^1\xi e^{(1-\xi )\text{ i }\tau \delta \kappa \triangle } f'(w(\kappa _n+\zeta \xi \tau \delta \kappa ))w'(\kappa _n+\zeta \xi \tau \delta \kappa ) d\zeta d\xi . \end{aligned} \end{aligned}$$

For the integrator (2.4), we have

$$\begin{aligned} \begin{aligned} \varPhi ^{\tau \delta \kappa }(w(\kappa _n)) =\,&C_{\tau }({\mathcal {W}})w(\kappa _n)+ \varepsilon \delta \kappa \int _{0}^{1}A_{\tau ,\sigma }({\mathcal {W}}) d\sigma f(w(\kappa _n))+ \delta \kappa ^2 C_1\\&+ \varepsilon \delta \kappa ^2 \int _{0}^1 \int _{0}^1\sigma A_{\tau ,\sigma }({\mathcal {W}}) f'(w(\kappa _n+\zeta \sigma \delta \kappa ))w'(\kappa _n+\zeta \sigma \delta \kappa ) d\zeta d\sigma \end{aligned} \end{aligned}$$

with $\left\| C_1\right\| _{H^{\alpha -4}}\lesssim 1$, where we replace $\varPhi ^{\sigma \delta \kappa }(w(\kappa _n))$ by $w(\kappa _n+\sigma \delta \kappa )$ in the numerical scheme and the error brought by this is denoted by $\delta \kappa ^2 C_1$. The combination of the above two equalities yields $\Vert \delta ^{n+\tau }\Vert _{H^{\alpha -4}}\lesssim \delta \kappa ^2$ for $0<\tau <1$, where the inequality $\left\| \int _{0}^{1}A_{\tau ,\sigma }({\mathcal {W}}) d\sigma -\tau \varphi _{1}(\tau {\mathcal {W}})\right\| _{H^{\alpha -4}}\lesssim \delta \kappa$ and the result of Lagrange interpolation have been used.

Then by the same arguments given above and by noticing $C_{1}({\mathcal {W}})=e^{\text{ i }\delta \kappa \triangle }$, the bound of $\Vert \delta ^{n+1}\Vert _{H^{\alpha -2}}$ can be derived.

Finally, in the light of

$$\begin{aligned} \begin{aligned} w(\kappa _{n+1}) =\,&e^{ \text{ i }\delta \kappa \triangle }w(\kappa _n)+ \varepsilon \delta \kappa \varphi _{1}( {\mathcal {W}})f(w(\kappa _n))+ \varepsilon \delta \kappa ^2 \varphi _{2}( {\mathcal {W}})f'(w(\kappa _n))w'(\kappa _n) \\&+\varepsilon \delta \kappa ^3 \int _{0}^1 \int _{0}^1(1-\zeta )\xi ^2 e^{(1-\xi ) \text{ i }\delta \kappa \triangle }\big ( f''(w(\kappa _n+\zeta \xi \delta \kappa ))(w'(\kappa _n+\zeta \xi \delta \kappa ))^2\\&+ f'(w(\kappa _n+\zeta \xi \delta \kappa ))w''(\kappa _n+\zeta \xi \delta \kappa )\big )d\zeta d\xi , \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \varPhi ^{ \delta \kappa }(w(\kappa _n)) =\,&e^{ \text{ i }\delta \kappa \triangle }w(\kappa _n)+ \varepsilon \delta \kappa \int _{0}^{1}A_{1,\sigma }({\mathcal {W}}) d\sigma f(w(\kappa _n))\\&+ \varepsilon \delta \kappa ^2 \int _{0}^{1}\sigma A_{1,\sigma }({\mathcal {W}}) d\sigma f'(w(\kappa _n))w'(\kappa _n) + \varepsilon \delta \kappa ^3 C_2\\&+\varepsilon \delta \kappa ^3 \int _{0}^1 \int _{0}^1(1-\zeta )\sigma ^2 A_{1,\sigma }({\mathcal {W}})\big ( f''(w(\kappa _n+\zeta \sigma \delta \kappa ))(w'(\kappa _n+\zeta \sigma \delta \kappa ))^2\\&+ f'(w(\kappa _n+\zeta \sigma \delta \kappa ))w''(\kappa _n+\zeta \sigma \delta \kappa )\big )d\zeta d\sigma , \end{aligned} \end{aligned}$$

with $\left\| C_2\right\| _{H^{\alpha -4}}\lesssim 1$, we obtain the bound of $\Vert \delta ^{n+1}\Vert _{H^{\alpha -4}}$ as follows

$$\begin{aligned} \begin{aligned}&\Vert \delta ^{n+1}\Vert _{H^{\alpha -4}} \lesssim&\sum \limits _{j=0}^{1}\varepsilon \delta \kappa ^{j+1}\left\| \varphi _{j+1}( {\mathcal {W}})-\int _{0}^{1}A_{1,\sigma }({\mathcal {W}})\dfrac{\sigma ^j}{j!}\mathrm{d}\sigma \right\| _{H^{\alpha -4}}+\varepsilon \delta \kappa ^3. \end{aligned} \end{aligned}$$

Using the results of $A_{1,\sigma }$:

$$\begin{aligned} \left\| \int _{0}^{1}A_{1,\sigma }({\mathcal {W}}) d\sigma - \varphi _{1}( {\mathcal {W}})\right\| _{H^{\alpha -4}}\lesssim 0, \ \ \left\| \int _{0}^{1}A_{1,\sigma }({\mathcal {W}}) \sigma d\sigma - \varphi _{2}( {\mathcal {W}})\right\| _{H^{\alpha -4}}\lesssim \delta \kappa , \end{aligned}$$

the last local error can be bounded. $\square$

Proof of Lemma 2

Employing the definition of the method, the isometry $C_{\tau }({\mathcal {W}})$ and the Lipschitz estimate of f, one gets

$$\begin{aligned} \begin{aligned}&\Vert \varPhi ^{\tau \delta \kappa }(v)-\varPhi ^{\tau \delta \kappa }(w)\Vert _{H^{\beta }}\le \Vert v-w\Vert _{H^{\beta }}+\varepsilon L C_A \int _{0}^{\delta \kappa } \Vert \varPhi ^{\sigma }(v)-\varPhi ^{\sigma }(w)\Vert _{H^{\beta }}d\sigma , \end{aligned} \end{aligned}$$

as long as $\varPhi ^{\sigma }(v),\ \varPhi ^{\sigma }(w)\in {\mathcal {H}}^{\alpha -2}_{R}$ for $\sigma \in [0,\delta \kappa ]$. Considering $\tau =1$ and using the Gronwall’s lemma yields

$$\begin{aligned} \Vert \varPhi ^{ \delta \kappa }(v)-\varPhi ^{ \delta \kappa }(w)\Vert _{H^{\beta }}\le e^{\varepsilon \delta \kappa L C_A}\Vert v-w\Vert _{H^{\beta }}, \end{aligned}$$

which gives the first statement of (3.7) by modifying $\delta \kappa$ to $\tau \delta \kappa$. Setting in particular $w = 0$ implies $\varPhi ^{\tau \delta \kappa }(v)\in {\mathcal {H}}^{\alpha -2}_{R}$ under the condition that $0<\delta \kappa <\delta \kappa _0$. It is also direct to have

$$\begin{aligned} \begin{aligned}&\Vert (\varPhi ^{ \delta \kappa }(v)-e^{\text{ i }\delta \kappa \triangle }v)-(\varPhi ^{ \delta \kappa }(w)-e^{\text{ i }\delta \kappa \triangle }w)\Vert _{H^{\beta }}\le \varepsilon \delta \kappa L C_A \Vert \varPhi ^{\tau \delta \kappa }(v)-\varPhi ^{\tau \delta \kappa }(w)\Vert _{H^{\beta }}. \end{aligned} \end{aligned}$$

The second result of (3.7) follows immediately from this inequality and the first statement. $\square$

Proof of Lemma 3

In order to derive the modulation equations for EP1, a new approach different from [19, 29, 30] is considered here. To this end, we define the operator $L^{k}$ and it can be expressed in Taylor expansions as follows:

$$\begin{aligned} \begin{aligned} L^{\langle j\rangle }_j=\,&\frac{1}{2} {\tilde{\epsilon }} h^2\omega _j\csc \big (\frac{1}{2}h\omega _j\big ) D+ \frac{1}{48} {\tilde{\epsilon }}^3 h^4\omega _j\csc \big (\frac{1}{2}h\omega _j\big ) D^3+\cdots ,\\ L^{k} = \,&\text{ i }h\varOmega \csc \big (\frac{1}{2}h\varOmega \big )\sin \big (\frac{1}{2}h(-\varOmega -(k \cdot \omega )I)\big ) \\&+ \frac{1}{2}{\tilde{\epsilon }} h^2 \varOmega \csc \big (\frac{1}{2}h\varOmega \big )\cos \big (\frac{1}{2}h((k \cdot \omega )I+\varOmega )\big )D +\cdots . \end{aligned} \end{aligned}$$

(6.1)

Moreover, for the operator $L_3^{k}(\sigma )$, we have

$$\begin{aligned} L_3^{k}(\frac{1}{2})=\cos \big (\frac{h(k \cdot \omega )}{2}\big )+\frac{1}{2}\sin \big (\frac{h(k \cdot \omega )}{2}\big )(\text{ i } h{\tilde{\epsilon }} D)+\cdots . \end{aligned}$$

By using the symmetry of the EP1 integrator and

$$\begin{aligned} \displaystyle \int _{0}^{1}f((1-\sigma )u^n+\sigma u^{n-1})d\sigma =\displaystyle \int _{0}^{1}f((1-\sigma )u^{n-1}+\sigma u^{n})d\sigma , \end{aligned}$$

we can rewrite the scheme of EP1 as³

$$\begin{aligned} \begin{aligned}&u^{n+1}-2\cos (h\varOmega )u^{n}+u^{n-1}\\&\quad =h\Big [\varphi _1(V)\displaystyle \int _{0}^{1}f((1-\sigma )u^{n}+\sigma u^{n+1})d\sigma -\varphi _1(-V) \displaystyle \int _{0}^{1}f((1-\sigma )u^{n-1}+\sigma u^{n})d\sigma \Big ]. \end{aligned} \end{aligned}$$

(6.2)

For the term $(1-\sigma )u^{n}+\sigma u^{n+1}$, we look for a modulated Fourier expansion of the form

$$\begin{aligned} \begin{aligned} {\tilde{u}}_h(t+\frac{h}{2},x,\sigma )= \sum \limits _{\left\| k\right\| \le K}w_{j(k)}^{k}\Big ({\tilde{\epsilon }}(t+\frac{h}{2}),\sigma \Big )\mathrm {e}^{\mathrm {i}(j(k) \cdot x)} \mathrm {e}^{-\mathrm {i}(k \cdot \omega ) (t+\frac{h}{2})}, \end{aligned} \end{aligned}$$

which leads to

$$\begin{aligned} \begin{aligned} w_{j(k)}^{k}\Big ({\tilde{\epsilon }}(t+\frac{h}{2}),\sigma \Big ) =&L^{k}_3(\sigma )z_{j(k)}^{k}\Big ({\tilde{\epsilon }}(t+\frac{h}{2})\Big ). \end{aligned} \end{aligned}$$

(6.3)

Likwise, for $(1-\sigma )u^{n-1}+\sigma u^{n}$, we have the following modulated Fourier expansion

$$\begin{aligned} \begin{aligned} {\tilde{u}}_h(t-\frac{h}{2},x,\sigma )= \sum \limits _{\left\| k\right\| \le K}w_{j(k)}^{k}\Big ({\tilde{\epsilon }}(t-\frac{h}{2}),\sigma \Big )\mathrm {e}^{\mathrm {i}(j(k) \cdot x)} \mathrm {e}^{-\mathrm {i}(k \cdot \omega )(t-\frac{h}{2})}. \end{aligned} \end{aligned}$$

Inserting (4.5) and (6.3) into (6.2) yields

$$\begin{aligned} \begin{aligned}&{\tilde{u}}(t+h,x)-2\cos (h\varOmega ){\tilde{u}}(t,x)+{\tilde{u}}(t-h,x)\\&\quad =h\Big [\varphi _1(V)\displaystyle \int _{0}^{1}f\big ({\tilde{u}}_{h}(t+\frac{h}{2},x,\sigma )\big )d\sigma -\varphi _1(-V) \displaystyle \int _{0}^{1}f\big ({\tilde{u}}_{h}(t-\frac{h}{2},x,\sigma )\big )d\sigma \Big ], \end{aligned} \end{aligned}$$

which can be expressed by operators as

$$\begin{aligned} \begin{aligned}&(\varphi _1(\text{ i }h \varOmega )\mathrm {e}^{\frac{1}{2} hD}-\varphi _1(-\text{ i }h \varOmega )\mathrm {e}^{-\frac{1}{2} hD})^{-1}( \mathrm {e}^{ hD}-2\cos (h\varOmega )+ \mathrm {e}^{- hD}){\tilde{u}}(t,x)\\&\quad =h\displaystyle \int _{0}^{1}f({\tilde{u}}_{h}(t,x,\sigma ))d\sigma . \end{aligned} \end{aligned}$$

(6.4)

On the other hand, we rewrite the nonlinearity f as:

$$\begin{aligned} \begin{aligned}&f(u) =-\text{ i }\sum \limits _{\left\| k\right\| \le K}\sum \limits _{j(k) \in {\mathcal {M}}}\sum \limits _{k^1+k^2-k^3=k }w^{k^1}_{l_1}w^{k^2}_{l_2}\overline{w^{k^3}_{l_3}} e^{\mathrm {i}(j(k)\cdot x)}\mathrm {e}^{-\mathrm {i}(k \cdot \omega ) t}, \end{aligned} \end{aligned}$$

where $j(k)=(j(k^1)+j(k^2)-j(k^3))\ \text{ mod }\ 2M$ if $k=k^1+k^2-k^3$. On the basis of this fact and (6.4), considering the jth Fourier coefficient and comparing the coefficients of $\mathrm {e}^{-\mathrm {i}(k\cdot \omega ) t}$, the result of this lemma is obtained. $\square$