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