Volume 37, Issue 1
Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation Without Loss of Regularity

Yongsheng Li, Yifei Wu & Fangyan Yao

Ann. Appl. Math., 37 (2021), pp. 1-21.

Published online: 2021-02

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  • Abstract

In this paper, we study the convergence rate of an Embedded exponential-type low-regularity integrator (ELRI) for  the  Korteweg-de Vries equation. We develop some new harmonic analysis techniques to handle the "stability" issue. In particular, we use a new stability estimate which allows us to avoid the use of the fractional Leibniz inequality,

image.png

and replace it by suitable inequalities without loss of regularity. Based on these techniques, we prove that the ELRI scheme proposed in [41] provides $\frac12$-order convergence accuracy  in $H^\gamma$ for any initial data belonging to $H^\gamma$ with $\gamma >\frac32$, which  does not require any additional derivative assumptions.

  • Keywords

The KdV equation, numerical solution, convergence analysis, error estimate, low regularity, fast Fourier transform.

  • AMS Subject Headings

65M12, 65M15, 35Q53

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yfy1357@126.com (Fangyan Yao)

  • BibTex
  • RIS
  • TXT
@Article{AAM-37-1, author = {Yongsheng and Li and and 18310 and and Yongsheng Li and Yifei and Wu and and 18311 and and Yifei Wu and Fangyan and Yao and yfy1357@126.com and 18312 and School of Mathematical Sciences, South China University ofTechnology, Guangzhou, Guangdong 510640, China and Fangyan Yao}, title = {Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation Without Loss of Regularity}, journal = {Annals of Applied Mathematics}, year = {2021}, volume = {37}, number = {1}, pages = {1--21}, abstract = {

In this paper, we study the convergence rate of an Embedded exponential-type low-regularity integrator (ELRI) for  the  Korteweg-de Vries equation. We develop some new harmonic analysis techniques to handle the "stability" issue. In particular, we use a new stability estimate which allows us to avoid the use of the fractional Leibniz inequality,

image.png

and replace it by suitable inequalities without loss of regularity. Based on these techniques, we prove that the ELRI scheme proposed in [41] provides $\frac12$-order convergence accuracy  in $H^\gamma$ for any initial data belonging to $H^\gamma$ with $\gamma >\frac32$, which  does not require any additional derivative assumptions.

}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2020-0001}, url = {http://global-sci.org/intro/article_detail/aam/18628.html} }
TY - JOUR T1 - Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation Without Loss of Regularity AU - Li , Yongsheng AU - Wu , Yifei AU - Yao , Fangyan JO - Annals of Applied Mathematics VL - 1 SP - 1 EP - 21 PY - 2021 DA - 2021/02 SN - 37 DO - http://doi.org/10.4208/aam.OA-2020-0001 UR - https://global-sci.org/intro/article_detail/aam/18628.html KW - The KdV equation, numerical solution, convergence analysis, error estimate, low regularity, fast Fourier transform. AB -

In this paper, we study the convergence rate of an Embedded exponential-type low-regularity integrator (ELRI) for  the  Korteweg-de Vries equation. We develop some new harmonic analysis techniques to handle the "stability" issue. In particular, we use a new stability estimate which allows us to avoid the use of the fractional Leibniz inequality,

image.png

and replace it by suitable inequalities without loss of regularity. Based on these techniques, we prove that the ELRI scheme proposed in [41] provides $\frac12$-order convergence accuracy  in $H^\gamma$ for any initial data belonging to $H^\gamma$ with $\gamma >\frac32$, which  does not require any additional derivative assumptions.

Yongsheng Li, Yifei Wu & Fangyan Yao. (1970). Convergence of an Embedded Exponential-Type Low-Regularity Integrators for the KdV Equation Without Loss of Regularity. Annals of Applied Mathematics. 37 (1). 1-21. doi:10.4208/aam.OA-2020-0001
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