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Volume 19, Issue 5
A Second-Order Embedded Low-Regularity Integrator for the Quadratic Nonlinear Schrödinger Equation on Torus

Fangyan Yao

Int. J. Numer. Anal. Mod., 19 (2022), pp. 656-668.

Published online: 2022-08

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  • Abstract

A new embedded low-regularity integrator is proposed for the quadratic nonlinear Schrödinger equation on the one-dimensional torus. Second-order convergence in $H^\gamma$ is proved for solutions in $C([0, T]; H^\gamma)$ with $\gamma > \frac{3}{2},$ i.e., no additional regularity in the solution is required. The proposed method is fully explicit and can be computed by the fast Fourier transform with $\mathcal{(O} log N)$ operations at every time level, where $N$ denotes the degrees of freedom in the spatial discretization. The method extends the first-order convergent low-regularity integrator in [14] to second-order time discretization in the case $\gamma >\frac{3}{2}$ without requiring additional regularity of the solution. Numerical experiments are presented to support the theoretical analysis by illustrating the convergence of the proposed method.

  • Keywords

Quadratic nonlinear Schrödinger equation, low-regularity integrator, second-order convergence, fast Fourier transform.

  • AMS Subject Headings

65M12, 65M15, 35Q55

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-656, author = {Yao , Fangyan}, title = {A Second-Order Embedded Low-Regularity Integrator for the Quadratic Nonlinear Schrödinger Equation on Torus}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {19}, number = {5}, pages = {656--668}, abstract = {

A new embedded low-regularity integrator is proposed for the quadratic nonlinear Schrödinger equation on the one-dimensional torus. Second-order convergence in $H^\gamma$ is proved for solutions in $C([0, T]; H^\gamma)$ with $\gamma > \frac{3}{2},$ i.e., no additional regularity in the solution is required. The proposed method is fully explicit and can be computed by the fast Fourier transform with $\mathcal{(O} log N)$ operations at every time level, where $N$ denotes the degrees of freedom in the spatial discretization. The method extends the first-order convergent low-regularity integrator in [14] to second-order time discretization in the case $\gamma >\frac{3}{2}$ without requiring additional regularity of the solution. Numerical experiments are presented to support the theoretical analysis by illustrating the convergence of the proposed method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/20931.html} }
TY - JOUR T1 - A Second-Order Embedded Low-Regularity Integrator for the Quadratic Nonlinear Schrödinger Equation on Torus AU - Yao , Fangyan JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 656 EP - 668 PY - 2022 DA - 2022/08 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/20931.html KW - Quadratic nonlinear Schrödinger equation, low-regularity integrator, second-order convergence, fast Fourier transform. AB -

A new embedded low-regularity integrator is proposed for the quadratic nonlinear Schrödinger equation on the one-dimensional torus. Second-order convergence in $H^\gamma$ is proved for solutions in $C([0, T]; H^\gamma)$ with $\gamma > \frac{3}{2},$ i.e., no additional regularity in the solution is required. The proposed method is fully explicit and can be computed by the fast Fourier transform with $\mathcal{(O} log N)$ operations at every time level, where $N$ denotes the degrees of freedom in the spatial discretization. The method extends the first-order convergent low-regularity integrator in [14] to second-order time discretization in the case $\gamma >\frac{3}{2}$ without requiring additional regularity of the solution. Numerical experiments are presented to support the theoretical analysis by illustrating the convergence of the proposed method.

Yao , Fangyan. (2022). A Second-Order Embedded Low-Regularity Integrator for the Quadratic Nonlinear Schrödinger Equation on Torus. International Journal of Numerical Analysis and Modeling. 19 (5). 656-668. doi:
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