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Local Discontinuous Galerkin Methods with Multistep Implicit–Explicit Time Discretization for Nonlinear Schrödinger Equations

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Abstract

In this paper, we investigate the local discontinuous Galerkin (LDG) methods coupled with multistep implicit–explicit (IMEX) time discretization to solve one-dimensional and two-dimensional nonlinear Schrödinger equations. In this approach, the nonlinear terms are treated explicitly, while the linear terms are handled implicitly. By the skew symmetry property of LDG operators and the properties of Gauss–Radau projections, we obtain error estimates for the prime and auxiliary variables, as well as the estimate for the time difference of the prime variables. These results, together with a carefully chosen numerical initial condition, allow us to obtain the optimal error estimate in both space and time for the fully discrete scheme. Numerical experiments are performed to verify the accuracy and efficiency of the proposed methods.

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Acknowledgements

The research of Y. Li was supported by National Key Research and Development Program of China 2021YFA1003004. The research of X. Zhong was partially supported by NSFC Grant 12272347. The authors would like to thank Prof. Qiang Zhang from Nanjing University for his valuable suggestions.

Funding

Author Y. Li was partially supported by National Key Research and Development Program of China 2021YFA1003004. Author X. Zhong was partially supported by NSFC Grant 12272347.

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Authors and Affiliations

  1. School of Computer Engineering and Science, Shanghai University, Shanghai, 200444, China

    Ying Li

  2. School of Computer Engineering and Science, Shanghai University, Shanghai, China

    Hui Shi

  3. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310058, Zhejiang, China

    Xinghui Zhong

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A The Expression of \(Z_{41}\)

A The Expression of \(Z_{41}\)

To simplify the presentation, the parameter \(\gamma \) is omitted in the following.

By (3.16), (2.17), and (3.11), we first rewrite the expression of \(F^nR^n-f^nr^n\) as

$$\begin{aligned} \begin{aligned} F^nR^n-f^nr^n&=F^n(R^n-r^n)+(F^n-f^n)r^n\\&=F^n(R^n-r^n)+r^n((R^n)^2-(r^n)^2+(S^n)^2-(s^n)^2)\\&=\left( (R^{n})^2+(S^{n})^2\right) e_r^n+r^n\left( (R^n+r^n)e_r^n+(S^n+s^n)e_s^n\right) \\&=\left( (R^{n})^2+(S^{n})^2+(r^n)^2+r^nR^n\right) e_r^n + r^n(s^n+S^n)e_s^n. \end{aligned} \end{aligned}$$
(A.1)

Similarly, \(F^{n-1}R^{n-1}-f^{n-1}r^{n-1}\) can be rewritten as

$$\begin{aligned} \begin{aligned}&F^{n-1}R^{n-1}-f^{n-1}r^{n-1}\\&=\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) e_r^{n-1} + r^{n-1}(s^{n-1}+S^{n-1})e_s^{n-1}. \end{aligned} \end{aligned}$$
(A.2)

Then, by subtracting (A.2) from (A.1), we have

$$\begin{aligned}&F^nR^n-f^nr^n-\left( F^{n-1}R^{n-1}-f^{n-1}r^{n-1}\right) \\&\quad =\left( (R^{n})^2+(S^{n})^2+(r^n)^2+r^nR^n\right) e_r^n -\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) e_r^{n-1}\\&\qquad + r^n(s^n+S^n)e_s^n-r^{n-1}(s^{n-1}+S^{n-1})e_s^{n-1}\\&\quad =\left( (R^{n})^2+(S^{n})^2+(r^n)^2+r^nR^n-(R^{n-1})^2-(S^{n-1})^2-(r^{n-1})^2-r^{n-1}R^{n-1}\right) e_r^n \\&\qquad +\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) (e_r^n-e_r^{n-1})\\&\qquad + \left( r^n(s^n+S^n)-r^{n-1}(s^{n-1}+S^{n-1})\right) e_s^n+r^{n-1}(s^{n-1}+S^{n-1})(e_s^n-e_s^{n-1})\\&\quad =(S^n+S^{n-1})(S^n-S^{n-1})e_r^n+(R^n+R^{n-1})(R^n-R^{n-1})e_r^n\\&\qquad +\left( (r^n-r^{n-1})(r^n+r^{n-1})+r^n(R^n-R^{n-1})+R^{n-1}(r^n-r^{n-1})\right) e_r^n \\&\qquad +\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) (e_r^n-e_r^{n-1})\\&\qquad + \left( r^n(S^n-S^{n-1})+S^{n-1}(r^n-r^{n-1})+r^n(s^n-s^{n-1})+s^{n-1}(r^n-r^{n-1})\right) e_s^n\\&\qquad +r^{n-1}(s^{n-1}+S^{n-1})(e_s^n-e_s^{n-1})\\&\quad =(S^n+S^{n-1})(S^n-S^{n-1})e_r^n+(R^n+R^{n-1}+r^n)\\&\qquad (R^n-R^{n-1})e_r^n+(r^n+r^{n-1}+R^{n-1})(r^n-r^{n-1})e_r^n \\&\qquad +\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) (e_r^n-e_r^{n-1})\\&\qquad + \left( r^n(S^n-S^{n-1})+(S^{n-1}+s^{n-1})(r^n-r^{n-1})+r^n(s^n-s^{n-1})\right) e_s^n\\&\qquad +r^{n-1}(s^{n-1}+S^{n-1})(e_s^n-e_s^{n-1})\\&\quad =(S^n+S^{n-1})(S^n-S^{n-1})e_r^n+(R^n+R^{n-1}+r^n)(R^n-R^{n-1})e_r^n\\&\qquad +(r^n+r^{n-1}+R^{n-1})(-(R^n-r^n)+(R^{n-1}-r^{n-1})+(R^n-R^{n-1})e_r^n \\&\qquad +\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) (e_r^n-e_r^{n-1})\\&\qquad + (r^n(S^n-S^{n-1})+(S^{n-1}+s^{n-1})(R^n-R^{n-1}-(R^n-r^n)+R^{n-1}-r^{n-1})\\&\qquad +r^n(S^n-S^{n-1}-(S^n-s^n)+S^{n-1}\\&\qquad -s^{n-1}))e_s^n+r^{n-1}(s^{n-1}+S^{n-1})(e_s^n-e_s^{n-1})\\&\quad =(S^n+S^{n-1})(S^n-S^{n-1})e_r^n\\&\qquad +(R^n+2R^{n-1}+2r^n+r^{n-1})(R^n-R^{n-1})e_r^n\\&\qquad -(r^n+r^{n-1}+R^{n-1})(e_r^n-e_r^{n-1})e_r^n\\&\qquad +\left( (R^{n-1})^2+(S^{n-1})^2+(r^{n-1})^2+r^{n-1}R^{n-1}\right) (e_r^n-e_r^{n-1})\\&\qquad +2r^n(S^n-S^{n-1})e_s^n+(S^{n-1}\\&\qquad +s^{n-1})(R^n-R^{n-1})e_s^n-(S^{n-1}+s^{n-1})(e_r^n-e_r^{n-1})e_s^n\\&\qquad -r^n(e_s^n-e_s^{n-1})e_s^n+r^{n-1}(s^{n-1}+S^{n-1})(e_s^n-e_s^{n-1})\\&\quad =((S^{n-1})^2+(R^{n-1})^2+r^{n-1}R^{n-1}+(r^{n-1})^2-e_r^n(R^{n-1}+r^n+r^{n-1})\\&\qquad -e_s^n(S^{n-1}+s^{n-1}))(e_r^{n}-e_r^{n-1})\\&\qquad +(r^{n-1}(S^{n-1}+s^{n-1})-r^{n}e_s^n)(e_s^n-e_s^{n-1})\\&\qquad +((S^n+S^{n-1})(S^n-S^{n-1})+(2r^n+r^{n-1}+2R^{n-1}+R^n)(R^n-R^{n-1}))e_r^n\\&\qquad +\left( 2r^n(S^{n}-S^{n-1})+(S^{n-1}+s^{n-1})(R^{n}-R^{n-1})\right) e_s^n, \end{aligned}$$

which completes the details of the expression for \(Z_{41}\).

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Li, Y., Shi, H. & Zhong, X. Local Discontinuous Galerkin Methods with Multistep Implicit–Explicit Time Discretization for Nonlinear Schrödinger Equations. J Sci Comput 101, 4 (2024). https://doi.org/10.1007/s10915-024-02647-6

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