Uniformly accurate nested Picard iterative integrators for the Klein-Gordon-Schrödinger equation in the nonrelativistic regime

Abstract

We establish a class of uniformly accurate nested Picard iterative integrator (NPI) Fourier pseudospectral methods for the nonlinear Klein-Gordon-Schrödinger equation (KGS) in the nonrelativistic regime, involving a dimensionless parameter ε ≪ 1 inversely proportional to the speed of light. Actually, the solution propagates waves in time with O(ε²) wavelength when 0 < ε ≪ 1, which brings significant difficulty in designing accurate and efficient numerical schemes. The NPI method is designed by separating the oscillatory part from the non-oscillatory part, and integrating the former exactly. Based on the Picard iteration, the NPI method can be applied to derive arbitrary higher-order methods in time with optimal and uniform accuracy (w.r.t. ε ∈ (0,1]), and the corresponding error estimates are rigorously established. In addition, the practical implementation of the second-order NPI method via Fourier pseupospectral discretization is clearly demonstrated, with extensions to the third-order NPI. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the proposed schemes.

Appendix A: Details of the third-order NPI method

Here, we shall give the details of programming by using MATLAB R2012a. At first, let

$$ \begin{array}{@{}rcl@{}} \psi_{\psi}^{n+1}&=&\psi_{\psi}^{n,3}:=e^{i{\Delta} \tau}\psi^{n}+\delta_{\psi}^{n,1}(\tau)+\delta_{\psi}^{n,2}(\tau)+\delta_{\psi}^{n,3}(\tau),\\ \phi_{\pm}^{n+1}&=&\phi_{\pm}^{n,3}:=e^{\mp i\mathcal{D}_{\varepsilon} \tau}\phi_{\pm}^{n}+\delta_{\pm}^{n,1}(\tau)+\delta_{\pm}^{n,2}(\tau)+\delta_{\pm}^{n,3}(\tau), \end{array} $$

can be stated as below by specifying $\delta _{\psi }^{n,3},\delta _{\pm }^{n,3}$, i.e. evaluating (2.22) for k = 3,

$$ \begin{array}{@{}rcl@{}} \delta_{\psi}^{n,3}&=&\sum\limits_{\pm}\bigg(\sum\limits_{k=0}^{2}p_{3,k,\pm}(s)\mathcal{F}^{n}_{3,k,\pm}+ \sum\limits_{k=5}^{6}p_{3,k,\pm}(s)\mathcal{F}^{n}_{3,k,\pm}+ \sum\limits_{k=0}^{3}p_{3,3,k,\pm}(s)\mathcal{F}^{n}_{3,3,k,\pm}\\ &&+\sum\limits_{k=0}^{1}p_{3,7,k,\pm}(s)\mathcal{F}^{n}_{3,7,k,\pm}+ \sum\limits_{k=0}^{5}p_{3,8,k,\pm}(s)\mathcal{F}^{n}_{3,8,k,\pm}\\&&+ \sum\limits_{k=0}^{4}\sum\limits_{\sigma=\pm}p_{3,k,\sigma,\pm}(s)\mathcal{F}^{n}_{3,k,\sigma,\pm}\bigg),\\ \delta_{\pm}^{n,3}&=&\sum\limits_{k=0}^{2}q_{3,k,\pm}(s)\mathcal{G}^{n}_{3,k,\pm}+ \sum\limits_{k=3}^{4}\sum\limits_{\sigma=\pm}q_{3,k,\sigma,\pm}(s)\mathcal{G}^{n}_{3,k,\sigma,\pm}\\ &&+\sum\limits_{k1=5}^{6}\sum\limits_{k2=0}^{4}\sum\limits_{\sigma=\pm}q_{3,k1,k2,\sigma,\pm}(s)\mathcal{G}^{n}_{3,k1,k2,\sigma,\pm}\\ &&+\sum\limits_{k=7}^{8}\sum\limits_{\sigma=\pm}q_{3,k,\sigma,\pm}(s)\mathcal{G}^{n}_{3,k,\sigma,\pm}+ \sum\limits_{\sigma_{1}=\pm}\sum\limits_{\sigma_{2}=\pm}q_{3,9,\sigma_{1},\sigma_{2},\pm}(s)\mathcal{G}^{n}_{3,9,\sigma_{1},\sigma_{2},\pm}, \end{array} $$

and the rest nonlinear terms are described below

$$ \begin{array}{@{}rcl@{}} \mathcal{F}^{n}_{3,0,\pm}&=&\pm i(\mathcal{B}_{2}\psi^{n})\mathcal{B}_{1}\phi^{n}_{\mp} +\frac{i}{2}\phi^{n}_{\mp}\mathcal{B}_{2}^{2}\psi^{n} +\frac{i}{2}\psi^{n}\mathcal{B}_{1}^{2}\phi_{\mp}^{n},\\ \mathcal{F}^{n}_{3,1,\pm}&=&\frac{i}{2}\mathcal{B}_{2}^{2}(\psi^{n}\phi^{n}_{\mp}), \mathcal{F}^{n}_{3,2,\pm}=i\mathcal{B}_{2}((\mathcal{B}_{2}\psi^{n})\phi_{\mp}^{n} \pm \psi^{n}\mathcal{B}_{1}\phi^{n}_{\mp}),\\ \mathcal{F}^{n}_{3,3,0,\pm}&=&i\mathcal{B}_{2}(\mathcal{F}^{n}_{1,+}\phi^{n}_{\mp}), \mathcal{F}^{n}_{3,3,1,\pm}=i\mathcal{B}_{2}(\mathcal{F}^{n}_{1,-}\phi^{n}_{\mp}),\\ \mathcal{F}^{n}_{3,5,\pm}&=&i\mathcal{B}_{2}(\mathcal{G}^{n}_{1,\mp}\psi^{n}), \mathcal{F}^{n}_{3,6,\pm}=i(\mathcal{B}_{2}\psi^{n})\mathcal{G}^{n}_{1,\mp},\\ \mathcal{F}^{n}_{3,3,2,\pm}&=&\pm i\mathcal{F}^{n}_{1,+}\mathcal{B}_{1}\phi^{n}_{\mp}, \mathcal{F}^{n}_{3,3,3 ,\pm}=\pm i\mathcal{F}^{n}_{1,-}\mathcal{B}_{1}\phi^{n}_{\mp}, \end{array} $$

$$ \begin{array}{@{}rcl@{}} \mathcal{F}^{n}_{3,k,\sigma,\pm}&=&i\mathcal{F}^{n}_{2,k,-\sigma}\phi_{\mp}^{n}, \mathcal{F}^{n}_{3,7,0,\pm}=i\mathcal{F}^{n}_{1,+}\mathcal{G}_{1,\mp}^{n},\\ \mathcal{F}^{n}_{3,7,1,\pm}&=&i\mathcal{F}^{n}_{1,-}\mathcal{G}_{1,\mp}^{n},\ \mathcal{F}^{n}_{3,8,k,\pm}=i\psi^{n}(\pm\mathcal{G}_{2,k}^{n}),\\ \mathcal{G}^{n}_{3,0,\pm}&=&\pm\mathcal{A}\left( \frac{1}{2}\psi^{n}\overline{\mathcal{B}_{2}^{2}\psi^{n}}+ \frac{1}{2}\overline{\psi^{n}}\mathcal{B}_{2}^{2}\psi^{n}+ |\mathcal{B}_{2}\psi^{n}|^{2} \right),\\ \mathcal{G}^{n}_{3,1,\pm}&=&\pm\frac{1}{2}\mathcal{A}\mathcal{B}_{1}^{2}|\psi^{n}|^{2}, \mathcal{G}^{n}_{3,2,\pm}=-\mathcal{A}\mathcal{B}_{1}\left( \psi^{n}\overline{\mathcal{B}_{2}\psi^{n}}+ \overline{\psi^{n}}\mathcal{B}_{2}\psi^{n} \right),\\ \mathcal{G}^{n}_{3,3,\sigma,\pm}&=&-\mathcal{A}\mathcal{B}_{1}(\overline{\psi^{n}}\mathcal{F}_{1,-\sigma}), \mathcal{G}^{n}_{3,4,\sigma,\pm}=-\mathcal{A}\mathcal{B}_{1}(\psi^{n}\overline{\mathcal{F}_{1,-\sigma}}),\\ \mathcal{G}^{n}_{3,5,k,\sigma,\pm}&=&\pm\mathcal{A}(\overline{\psi^{n}}\mathcal{F}_{2,k,-\sigma}), \mathcal{G}^{n}_{3,6,k,\sigma,\pm}=\pm\mathcal{A}(\psi^{n}\overline{\mathcal{F}_{2,k,-\sigma}}),\\ \mathcal{G}^{n}_{3,7,\sigma,\pm}&=&\pm\mathcal{A}(\mathcal{F}_{1,-\sigma}\overline{\mathcal{B}_{2}\psi^{n}}), \mathcal{G}^{n}_{3,8,\sigma,\pm}=\pm\mathcal{A}(\overline{\mathcal{F}_{1,-\sigma}}\mathcal{B}_{2}\psi^{n}),\\ \mathcal{G}^{n}_{3,9,\sigma_{1},\sigma_{2},\pm}&=&\pm\mathcal{A}(\mathcal{F}_{1,-\sigma_{1}}\overline{\mathcal{F}_{1,-\sigma_{2}}}), \end{array} $$

Finally, the coefficients are given by

$$ \begin{array}{@{}rcl@{}} p_{3,0,\pm}(s)&=&{{\int}_{0}^{s}}e^{\pm iw/\varepsilon^{2}}w^{2} dw, p_{3,1,\pm}(s)={{\int}_{0}^{s}}e^{\pm iw/\varepsilon^{2}}(s-w)^{2} dw,\\ p_{3,2,\pm}(s)&=&{{\int}_{0}^{s}}e^{\pm iw/\varepsilon^{2}}w(s-w) dw,\\ p_{3,3,k,\pm}(s)&=&{{\int}_{0}^{s}}e^{\pm iw/\varepsilon^{2}}p_{1,k}(w)(s-w) dw,\\ p_{3,3,k+2,\pm}(s)&=&{{\int}_{0}^{s}}e^{\pm iw/\varepsilon^{2}}p_{1,k}(w)w dw, k=0,1,\\ p_{3,5,\pm}(s)&=&{{\int}_{0}^{s}}q_{1,\pm}(w)(s-w) dw, p_{3,6,\pm}(s)={{\int}_{0}^{s}}q_{1,\pm}(w)w dw,\\ p_{3,7,k,\pm}(s)&=&{{\int}_{0}^{s}}p_{1,k}(w)q_{1,\pm}(w)dw, k=0,1,\\ p_{3,k,\sigma,\pm}&=&{{\int}_{0}^{s}}e^{\pm iw/\varepsilon^{2}}p_{2,k,\sigma}(w)dw,k=0,1,\dots,4,\sigma=\pm,\\ p_{3,8,k,\pm}&=&{{\int}_{0}^{s}}q_{2,\pm,k}(w)dw,k=0,1,\dots,5,\\ q_{3,0,\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}w^{2} dw, q_{3,1,\pm}(s)={{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}(s-w)^{2} dw,\\ q_{3,2,\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}w(s-w) dw,\\ q_{3,3,\sigma,\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}p_{1,\sigma}(w)(s-w) dw,\\ q_{3,4,\sigma,\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}\overline{p_{1,\sigma}}(w)(s-w) dw, \end{array} $$

$$ \begin{array}{@{}rcl@{}} q_{3,5,k,\sigma,\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}p_{2,k,\sigma}(w) dw, q_{3,6,k,\sigma,\pm}(s)={{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}\overline{p_{2,k,\sigma}}(w) dw,\\ q_{3,7,\sigma,\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}p_{1,\sigma}(w)w dw, q_{3,8,\sigma,\pm}(s)={{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}\overline{p_{1,\sigma}}(w)w dw,\\ q_{3,9,\sigma_{1},\sigma_{2},\pm}(s)&=&{{\int}_{0}^{s}}e^{\mp i(s-w)/\varepsilon^{2}}p_{1,\sigma_{1}}(w)\overline{p_{1,\sigma_{2}}}(w) dw. \end{array} $$