A Semi-discrete First-Order Low Regularity Exponential Integrator for the “good” Boussinesq Equation Without Loss of Regularity

Abstract

In this paper, we propose a semi-discrete first-order low regularity exponential-type integrator (LREI) for the “good” Boussinesq equation. It is shown that the method is convergent linearly in the space $H^r$ for solutions belonging to $H^{r+p(r)}$ where $0\le p(r)\le 1$ is non-increasing with respect to r, which means less additional derivatives might be needed when the numerical solution is measured in a more regular space. Particularly, the LREI presents the first-order accuracy in $H^{r}$ with no assumptions of additional derivatives when $r>5/2$. This is the first time to propose a low regularity method which achieves the optimal first-order accuracy without loss of regularity for the GB equation. The convergence is confirmed by extensive numerical experiments.

Appendix

In this section, we present a first-order operator splitting scheme based on the equivalent form (3.3) of the GB equation. The operator splitting methods for the time integration of (3.3) are based on the splitting

$$\begin{aligned} \partial _{t}u =X_1(u)+X_2(u), \end{aligned}$$

where $X_1 (u)=i\langle \partial _x^2 \rangle u, \quad X_2(u)=-iB\left( \frac{1}{4}(u+ \overline{u})^2+(at+b)(u + \overline{u})\right)$, and the solutions of the subproblems

$$\begin{aligned} \left\{ \begin{aligned}&\partial _{t}\nu (t,x)=X_1(\nu (t,x)),\\&\nu (0,x)=\nu _0(x), \quad x\in \mathbb {T}, \quad t>0, \end{aligned}\right. \quad \left\{ \begin{aligned}&\partial _{t}\omega (t,x) =X_2(\omega (t,x)),\\&\omega (0,x)=\omega _0(x), \quad x\in \mathbb {T}, \quad t>0. \end{aligned}\right. \end{aligned}$$

For the linear subproblem, the associated evolution operator is given by

$$\begin{aligned} \nu (t, \cdot )=\varPhi _{X_1}^t(\nu _0)=\textrm{e}^{it\langle \partial _x^2 \rangle }\nu _0. \end{aligned}$$

To integrate the nonlinear subequation, writing $\omega =\omega _1+i\omega _2$ with $\omega _1, \omega _2\in \mathbb {R}$, we are led to

$$\begin{aligned} i\partial _t(\omega _1+i\omega _2)=B\left[ \omega _1^2+2(at+b)\omega _1\right] , \end{aligned}$$

which implies

$$\begin{aligned} \partial _t\omega _1= 0,\quad -\partial _t\omega _2=B\left[ \omega _1^2+2(at+b)\omega _1\right] . \end{aligned}$$

Thus one obtains

$$\begin{aligned} \omega _1(t)=\omega _1(0):=\omega _1^0,\quad -\partial _t\omega _2=B\left[ (\omega _1^0)^2+2(at+b)\omega _1^0\right] , \end{aligned}$$

which gives that

$$\begin{aligned} \omega _2(t)=\omega _2^0-Bt(\omega _1^0)^2-aBt^2\omega _1^0-2bBt\omega _1^0, \quad \omega _2^0:=\omega _2(0). \end{aligned}$$

Hence we get

$$\begin{aligned} \omega (t)&=\omega _1(t)+i\omega _2(t)=\omega _1^0+i\left[ \omega _2^0-Bt(\omega _1^0)^2- aBt^2\omega _1^0-2bBt\omega _1^0\right] \\&=\omega _0-itB\left[ \frac{1}{4}\left( \omega _0+\overline{\omega _0}\right) ^2+ (\frac{at}{2}+b)\left( \omega _0+\overline{\omega _0}\right) \right] \\&:=\varPhi _{X_2}^t(\omega _0). \end{aligned}$$

Hence, the first-order Lie-Trotter splitting scheme reads as

$$\begin{aligned} u^{n+1}=\varPhi ^{\tau }(u^n)=\varPhi ^{\tau }_{X_1}(\varPhi ^{\tau }_{X_2}(u^n)), \quad u^0=u(0,x), \end{aligned}$$

and the numerical solution $z^n$ and $z_t^n$ can be recovered by (3.26). Furthermore, high-order splitting methods can be constructed correspondingly.