Uniformly Accurate Nested Picard Iterative Integrators for the Klein-Gordon Equation in the Nonrelativistic Regime

Abstract

In this paper, a class of uniformly accurate nested Picard iterative integrator (NPI) Fourier pseudospectral methods is proposed for the nonlinear Klein-Gordon equation (NLKG) in the nonrelativistic regime, involving a dimensionless parameter $\varepsilon \ll 1$ inversely proportional to the speed of light. For $0<\varepsilon \ll 1$, the solution propagates waves in time with $O(\varepsilon ^2)$ wavelength, which brings significant difficulty in designing accurate and efficient numerical schemes. The idea of NPI methods can be applied to derive arbitrary higher-order methods in time with optimal and uniform accuracy (w.r.t. $\varepsilon \in (0,1]$). Detailed constructions of the NPI methods up to the third order in time are presented for NLKG with a cubic/quadratic nonlinear term, where the corresponding error estimates are rigorously analyzed. In addition, the practical implementation of the second-order NPI method via Fourier pseupospectral discretization is clearly demonstrated. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the proposed schemes.

Appendices

Appendix

Appendix A. Details of the Third-Order NPI Method

Here, we shall give the details of programming by using Matlab R2012a. At first, let $\psi _{\pm }^{n+1}=\psi _{\pm }^{n,3}:=e^{\mp i\mathcal {D}_\varepsilon \tau }\psi _\pm ^{n}+\delta _{\pm }^{n,1}(\tau )+\delta _{\pm }^{n,2}(\tau )+\delta _{\pm }^{n,3}(\tau )$, can be stated as below by specifying $\delta _{\pm }^{n,3}$, i.e. evaluating (2.22) for $k=3$,

$$\begin{aligned}&\delta _{\pm }^{n,3}(s)\\&\quad =s^2p_{-2}(\pm s)\frac{1}{2}\mathcal {B}^2\mathcal {F}_{1,\pm }^n+ s^2p_0(\pm s)\frac{1}{2}\mathcal {B}^2\mathcal {F}_{2,\pm }^n + s^2p_{2}(\pm s)\frac{1}{2}\mathcal {B}^2\mathcal {F}_{2,\mp }^n+ s^2p_{4}(\pm s)\frac{1}{2}\mathcal {B}^2\mathcal {F}_{1,\mp }^n\\&\qquad +\sum _{k=1}^6\sum \limits _{\sigma =\pm }[( \mp sq_{k,\sigma }(\pm s) +q_{k,1,\sigma }(\pm s) )\mathcal {B}\mathcal {F}_{2,k,\pm \sigma }^n\,\pm q_{k,1,\sigma }(\pm s)\mathcal {F}_{3,k,\pm \sigma }^n]\\&\qquad +p_{-2,2}(\pm s)( 1/2\mathcal {B}^2\mathcal {F}_{1,\pm }^n \pm \mathcal {B}\mathcal {G}_{3,\pm }^n +\mathcal {G}_{1,3,\pm }^n +\mathcal {E}_{1,\pm }^n)\\&\qquad +p_{0,2}(\pm s)( 1/2\mathcal {B}^2\mathcal {F}_{2,\pm }^n \pm \mathcal {B}\mathcal {G}_{4,\pm }^n +\mathcal {G}_{2,3,\pm }^n +\mathcal {E}_{2,\pm }^n )\\&\qquad +p_{2,2}(\pm s)( 1/2\mathcal {B}^2\mathcal {F}_{2,\mp }^n \pm \mathcal {B}\mathcal {G}_{4,\mp }^n +\mathcal {G}_{2,3,\mp }^n +\mathcal {E}_{2,\mp }^n )\\&\qquad +p_{4,2}(\pm s)( 1/2\mathcal {B}^2\mathcal {F}_{1,\mp }^n \pm \mathcal {B}\mathcal {G}_{3,\mp }^n +\mathcal {G}_{1,3,\mp }^n +\mathcal {E}_{1,\mp }^n )\\&\qquad +p_{-2,1}(\pm s)( \mp s\mathcal {B}^2\mathcal {F}_{1,\pm }^n -s\mathcal {B}\mathcal {G}_{3,\pm }^n ) +p_{0,1}(\pm s)( \mp s\mathcal {B}^2\mathcal {F}_{2,\pm }^n -s\mathcal {B}\mathcal {G}_{4,\pm }^n )\\&\qquad +p_{2,1}(\pm s)( \mp s\mathcal {B}^2\mathcal {F}_{2,\mp }^n -s\mathcal {B}\mathcal {G}_{4,\mp }^n ) +p_{4,1}(\pm s)( \mp s\mathcal {B}^2\mathcal {F}_{1,\mp }^n -s\mathcal {B}\mathcal {G}_{3,\mp }^n )\\&\qquad +\sum _{k=1}^4\sum \limits _{\sigma _1=\pm }\sum \limits _{\sigma _2=\pm }(r_{k,\sigma _1,\sigma _2}(\pm s)\mathcal {H}^n_{k,\pm ,\pm \sigma _1,\pm \sigma _2} + r_{k+4,\sigma _1,\sigma _2}(\pm s)\mathcal {H}^n_{k,\mp ,\pm \sigma _1,\pm \sigma _2})\nonumber \\&\qquad + \sum _{k=1}^6\sum \limits _{\sigma =\pm }[q_{k,2,\sigma }(\pm s)\mathcal {F}_{4,k,\pm \sigma }^n \pm q_{k,4,\sigma }(\pm s)\mathcal {F}_{4,k+6,\pm \sigma }^n +q_{k,3,\sigma }(\pm s)\mathcal {F}_{5,k,\pm \sigma }^n]\\&\qquad + \sum _{j=1}^{3}\sum _{k=1}^{6}\sum \limits _{\sigma =\pm } m_{j,k,\sigma }(\pm s)\mathcal {F}_{6,j,k,\pm \sigma }^n, \end{aligned}$$

where for $m=1,2$ and $\sigma _1,\sigma _2=\pm$,

$$\begin{aligned}&\mathcal {G}_{1,3,\pm }^n=\mathcal {A} \left( \frac{1}{2}(\psi _\pm ^n)^2\overline{\mathcal {B}^2\psi _\mp ^n}+ (\mathcal {B}^2\psi _\pm ^n)\psi _\pm ^n\overline{\psi _\mp ^n}\right) ,\\&\mathcal {G}_{2,3,\pm }^n=\mathcal {A} \bigg (\psi _\pm ^n\overline{\psi _\mp ^n}(\mathcal {B}^2\psi _\mp ^n)+(|\psi _\pm ^n|^2+|\psi _{\mp }^n|^2)(\mathcal {B}^2\psi _\pm ^n)\\&\qquad \qquad + \frac{1}{2}(\psi _\pm ^n)^2\overline{\mathcal {B}^2\psi _\pm ^n}+\psi _\pm ^n\psi _\mp ^n\overline{(\mathcal {B}^2\psi _\mp ^n)}\bigg ) ,\\&\mathcal {F}_{3,m,\pm }^n=\mathcal {A}\left( 2(\mp \psi _{\mp }^n\overline{\mathcal {B}\psi _{\pm }^n} \pm \overline{\psi _{\pm }^n}\mathcal {B}\psi _{\mp }^n)\mathcal {F}_{m,\mp }^n -2(\pm \psi _{\mp }^n\mathcal {B}\psi _{\mp }^n)\overline{\mathcal {F}_{m,\pm }^n}\right) ,\\&\mathcal {F}_{3,m+2,\pm }^n=\mathcal {A}\bigg (2(\psi _-^n\overline{\mathcal {B}\psi _-^n}- \psi _+^n\overline{\mathcal {B}\psi _+^n} - \overline{\psi }_+^n\mathcal {B}\psi _+^n+ \overline{\psi }_-^n\mathcal {B}\psi _-^n)\mathcal {F}_{m,\mp }^n\\&\qquad \qquad - 2(\psi _+^n\mathcal {B}\psi _-^n- \psi _-^n\mathcal {B}\psi _+^n)\overline{\mathcal {F}_{m,\pm }^n}\bigg ),\\&\mathcal {F}_{3,m+4,\pm }^n=\mathcal {A}\left( 2(\pm \psi _{\pm }^n\overline{\mathcal {B}\psi _{\mp }^n} \mp \overline{\psi _{\mp }^n}\mathcal {B}\psi _{\pm }^n)\mathcal {F}_{m,\mp }^n -2(\mp \psi _{\pm }^n\mathcal {B}\psi _{\pm }^n)\overline{\mathcal {F}_{m,\pm }^n}\right) ,\\&\mathcal {E}_{1,\pm }^n=\mathcal {A}(-2\psi _{\pm }^n(\mathcal {B}\psi _{\pm }^n)\overline{\mathcal {B}\psi _{\mp }^n} + \overline{\psi _{\mp }^n}(\mathcal {B}\psi _{\pm }^n)^2),\\&\mathcal {E}_{2,\pm }^n=\mathcal {A}\bigg ( 2\psi _{\pm }^n|\mathcal {B}\psi _{\mp }^n|^2+2\psi _{\pm }^n|\mathcal {B}\psi _{\pm }^n|^2- 2\psi _{\mp }^n(\mathcal {B}\psi _{\pm }^n)\overline{\mathcal {B}\psi _{\mp }^n}\\&\qquad \qquad -2\overline{\psi _{\mp }^n}(\mathcal {B}\psi _{\pm }^n)\mathcal {B}\psi _{\mp }^n +\overline{\psi _{\pm }^n}(\mathcal {B}\psi _{\pm }^n)^2 \bigg ),\\&\mathcal {H}^n_{1,\pm ,\sigma _1,\sigma _2}= \mathcal {A}(-2\psi _{\pm }^nF_{1,\sigma _1}^n\overline{F_{1,-\sigma _2}^n}+ \overline{\psi _{\mp }^n}F_{1,\sigma _1}^nF_{1,\sigma _2}^n),\\&\mathcal {H}^n_{2,\pm ,\sigma _1,\sigma _2}=\mathcal {A}(-2\psi _{\pm }^nF_{1,\sigma _1}^n\overline{F_{2,-\sigma _2}^n}+ \overline{\psi _{\mp }^n}F_{1,\sigma _1}^nF_{2,\sigma _2}^n),\\&\mathcal {H}^n_{3,\pm ,\sigma _1,\sigma _2}= \mathcal {A}(-2\psi _{\pm }^nF_{2,\sigma _1}^n\overline{F_{1,-\sigma _2}^n}+ \overline{\psi _{\mp }^n}F_{2,\sigma _1}^nF_{1,\sigma _2}^n),\\&\mathcal {H}^n_{4,\pm ,\sigma _1,\sigma _2}= \mathcal {A}(-2\psi _{\pm }^nF_{2,\sigma _1}^n\overline{F_{2,-\sigma _2}^n}+ \overline{\psi _{\mp }^n}F_{2,\sigma _1}^nF_{2,\sigma _2}^n), \end{aligned}$$

and the rest nonlinear terms are described below

$$\begin{aligned}&\mathcal {F}_{4,m,\pm }^n= \mathcal {A}\left( 2\psi _{\mp }^n\overline{\psi _{\pm }^n}\mathcal {G}_{m,\mp }^n+(\psi _\mp ^n)^2\overline{\mathcal {G}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{4,m+2,\pm }^n= \mathcal {A}\left( 2(|\psi _+^n|^2+|\psi _-^n|^2)\mathcal {G}_{m,\mp }^n+ 2\psi _+^n\psi _-^n\overline{\mathcal {G}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{4,m+4,\pm }^n= \mathcal {A}\left( 2\psi _{\pm }^n\overline{\psi _{\mp }^n}\mathcal {G}_{m,\mp }^n+(\psi _\pm ^n)^2\overline{\mathcal {G}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{4,m+6,\pm }^n= \mathcal {A}\left( 2\psi _{\mp }^n\overline{\psi _{\pm }^n}\mathcal {G}_{m,\mp }^n- (\psi _\mp ^n)^2\overline{\mathcal {G}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{4,m+8,\pm }^n= \mathcal {A}\left( 2(|\psi _+^n|^2+|\psi _-^n|^2)\mathcal {G}_{m,\mp }^n- 2\psi _+^n\psi _-^n\overline{\mathcal {G}_{m,\pm }^n}\right) , m=1,2,\\&\mathcal {F}_{4,m+10,\pm }^n= \mathcal {A}\left( 2\psi _{\pm }^n\overline{\psi _{\mp }^n}\mathcal {G}_{m,\mp }^n- (\psi _\pm ^n)^2\overline{\mathcal {G}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{5,m,\pm }^n= \mathcal {A}\left( 2\psi _{\mp }^n\overline{\psi _{\pm }^n}\mathcal {B}\mathcal {F}_{m,\mp }^n+ (\psi _\mp ^n)^2\overline{\mathcal {B}\mathcal {F}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{5,m+2,\pm }^n=\mathcal {A}\left( 2(|\psi _+^n|^2+|\psi _-^n|^2)\mathcal {B}\mathcal {F}_{m,\mp }^n+ 2\psi _+^n\psi _-^n\overline{\mathcal {B}\mathcal {F}_{m,\pm }^n}\right) ,m=1,2,\\&\mathcal {F}_{5,m+4,\pm }^n= \mathcal {A}\left( 2\psi _{\pm }^n\overline{\psi _{\mp }^n}\mathcal {B}\mathcal {F}_{m,\mp }^n+ (\psi _\pm ^n)^2\overline{\mathcal {B}\mathcal {F}_{m,\pm }^n}\right) ,\quad m=1,2,\\&\mathcal {F}_{6,1,k,\pm }^n=\mathcal {A}\left( 2(|\psi _+^n|^2+|\psi _-^n|^2)\mathcal {F}_{2,k,\pm }^n+ 2\psi _+^n\psi _-^n\overline{\mathcal {F}_{2,k,\mp }^n}\right) , k=1,2,\dots ,6,\\&\mathcal {F}_{6,2,k,\pm }^n= \mathcal {A}\left( 2\psi _{\mp }^n\overline{\psi _{\pm }^n}\mathcal {F}_{2,k,\pm }^n+(\psi _\mp ^n)^2\overline{\mathcal {F}_{2,k,\mp }^n}\right) ,\quad k=1,2,\dots ,6,\\&\mathcal {F}_{6,3,k,\pm }^n= \mathcal {A}\left( 2\psi _{\pm }^n\overline{\psi _{\mp }^n}\mathcal {F}_{2,k,\pm }^n+(\psi _\pm ^n)^2\overline{\mathcal {F}_{2,k,\mp }^n}\right) ,\quad k=1,2,\dots ,6. \end{aligned}$$

Finally, the coefficients are given by $p_{0,2}(s)=e^{\frac{-is}{\varepsilon ^2}}\frac{s^3}{3}$, $p_{\pm 2,2}(s)=e^{\frac{-is}{\varepsilon ^2}}\int _{0}^{s}w^2e^{\frac{\pm 2iw}{\varepsilon ^2}}dw$, $p_{4,2}(s)=e^{\frac{-is}{\varepsilon ^2}}\int _{0}^{s}w^2e^{\frac{4 iw}{\varepsilon ^2}}dw$ and

$$\begin{aligned}&q_{1+k,1,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}} e^{\frac{\pm (2-k) iw}{\varepsilon ^2}}wg_{1,1}(\pm w)\,dw,\,k=0,2,4,\\&q_{2+k,1,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}} wg_{2,1}(\pm w)\,dw,k=0,2,4,\\&g_{1,1}(\pm w)=p_{-2}(\mp w) + p_4(\pm w),\,g_{2,1}(\pm w)= p_{2}(\mp w) + p_0(\pm w),\\&q_{1+k,2,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}}g_{1,2}(\pm w) \,dw,\,k=0,2,4,\\&q_{2+k,2,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}} g_{2,2}(\pm w)\,dw,\,k=0,2,4,\\&g_{1,2}(\pm w)=p_{-2,1}(\mp w)+p_{4,1}(\pm w),\,g_{2,2}(\pm w)=p_{2,1}(\pm w)+p_{0,1}(\mp w),\\&q_{1+k,3,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}} g_{1,3}(\pm w)\,dw,\,k=0,2,4,\\&q_{2+k,3,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}} g_{2,3}(\pm w)\,dw,\,k=0,2,4,\\&g_{1,3}(\pm w)=w(\pm p_{-2}(\mp w) \mp p_4(\pm w)),\,g_{2,3}(\pm w)=w(\mp p_2(\pm w) \pm p_0(\mp w)),\\&q_{1+k,4,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}}g_{1,4}(\pm w)\,dw,\,k=0,2,4,\\&q_{2+k,4,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}}e^{\frac{\pm (2-k) iw}{\varepsilon ^2}} g_{1,4}(\pm w)\,dw,\,k=0,2,4,\\&g_{1,4}(\pm w)=\mp p_{-2,1}(\mp w) \pm p_{4,1}(\pm w),\,g_{2,4}(\pm w)=\pm p_{2,1}(\pm w)\mp p_{0,1}(\mp w),\\&r_{1+m,\sigma _1,\sigma _2}(s)=e^{\frac{-is}{\varepsilon ^2}}\int _0^s g_{1+\frac{m}{2},1}(\sigma _1 w)g_{1,1}(\sigma _2 w)\,dw,\quad m=0,2\\&r_{2+m,\sigma _1,\sigma _2}(s)=e^{\frac{-is}{\varepsilon ^2}}\int _0^s g_{1+\frac{m}{2},1}(\sigma _1 w)g_{2,1}(\sigma _2 w)\,dw,\quad m=0,2\\&r_{5+m,\sigma _1,\sigma _2}(s)=e^{\frac{-is}{\varepsilon ^2}}\int _0^s e^{\frac{2iw}{\varepsilon ^2}}g_{1+\frac{m}{2},1}(\sigma _1 w)g_{1,1}(\sigma _2 w)\,dw,\quad m=0,2,\\&r_{6+m,\sigma _1,\sigma _2}(s)=e^{\frac{-is}{\varepsilon ^2}}\int _0^se^{\frac{2iw}{\varepsilon ^2}} g_{1+\frac{m}{2},1}(\sigma _1 w)g_{2,1}(\sigma _2 w)\,dw,\quad m=0,2,\\&m_{1,k,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}} g_{k,6}(w)\,dw,\, m_{2,k,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}} e^{\frac{\pm 2iw}{\varepsilon ^2}}g_{k,6}(w)\,dw,\\&m_{3,k,\pm }(s)=\int _0^se^{\frac{- i(s-w)}{\varepsilon ^2}} e^{\frac{\mp 2iw}{\varepsilon ^2}}g_{k,6}(w)\,dw,\, g_{k,6}(w)=q_{k,\pm }( w)+q_{k,\mp }(- w),\,1\le k\le 6. \end{aligned}$$