Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (4): 1536-1549.doi: 10.1007/s10473-024-0419-1

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A LOW-REGULARITY FOURIER INTEGRATOR FOR THE DAVEY-STEWARTSON II SYSTEM WITH ALMOST MASS CONSERVATION

Cui Ning1, Chenxi Hao2, Yaohong Wang2,*   

  1. 1. School of Financial Mathematics and Statistics, Guangdong University of Finance, Guangzhou 510521, China;
    2. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
  • Received:2022-11-30 Revised:2023-06-08 Online:2024-08-25 Published:2024-08-30
  • Contact: *E-mail: yaohong@tju.edu.cn
  • About author:E-mail: cuiningmath@sina.com; haochenxi2015@126.com
  • Supported by:
    Ning's work was supported by the NSFC (11901120), the Science and Technology Program of Guangzhou, China (2024A04J4027); Hao's work was supported by the NSFC (12171356).

Abstract:

In this work, we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-Stewartson II system (hyperbolic-elliptic case). Arbitrary order mass convergence could be achieved by the suitable addition of correction terms, while keeping the first order accuracy in $H^{\gamma}\times H^{\gamma+1}$ for initial data in $H^{\gamma+1}\times H^{\gamma+1}$ with $\gamma>1$. The main theorem is that, up to some fixed time $T$, there exist constants $\tau_0$ and $C$ depending only on $T$ and $\|u\|_{L^{\infty}\left((0, T) ; H^{\gamma+1}\right)}$ such that, for any $0<\tau\leq\tau_0$, we have that

$\begin{equation*}\left\|u\left(t_{n}, \cdot\right)-u^{n}\right\|_{H^{\gamma}} \leq C \tau,\quad \left\|v\left(t_{n}, \cdot\right)-v^{n}\right\|_{H^{\gamma+1}} \leq C \tau, \end{equation*}$

where $u^n$ and $v^n$ denote the numerical solutions at $t_n=n\tau$. Moreover, the mass of the numerical solution $M(u^n)$ satisfies that

$\begin{equation*}\left|M\left(u^{n}\right)-M\left(u_{0}\right)\right| \leq C \tau^{5}.\end{equation*}$

Key words: Davey-Stewartson II system, low-regularity, exponential integrator, first order accuracy, mass conservation

CLC Number: 

  • 65M12
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