Numerical simulation of solitary waves of Rosenau–KdV equation by Crank–Nicolson meshless spectral interpolation method

Abstract

An efficient and accurate Crank–Nicolson meshless spectral radial point interpolation (CN-MSRPI) method is proposed for the numerical solution of nonlinear Rosenau–KdV equation. The proposed method uses meshless shape functions, owing Kronecker delta property, for approximation of spatial operator. Crank–Nicolson difference scheme is used for temporal operator approximation. Single solitary wave motion, interaction of double and triple solitary waves as well as generation of train of solitary waves from initial data are numerically simulated. Error analysis is made via computation of discrete $L_{\infty }$, $L_{2}$ and $L_{\mathrm{rms}}$ error norms. Efficiency of the proposed numerical scheme is assessed via variation of number of nodes N and time step-size $\tau$. Two invariant quantities correspond to mass and energy are computed using the proposed method for further validation. Stability of the proposed method is discussed and verified computationally. Comparison of obtained results made with exact and existing results in the literature revealed the proposed CN-MSRPI method superiority.

Appendix A

In order to show linearization process for the nonlinear term $UU_{x}$ in Eq. (6) let us consider

$$\begin{aligned} \bigg [U(x)U_{x}(x)\bigg ]^{n+1}=[UU_{x}](t^{n}+\tau ,x) \end{aligned}$$

(16)

where $\tau >0$ is the time step-size. By expanding R.H.S. of Eq. (16) using Taylor’s series about $\tau$, one can write

$$\begin{aligned}{}[UU_{x}](x,t^{n}+\tau )= & {} UU_{x}(x,t^{n})+\tau \bigg [\partial _{t}(UU_{x})\bigg ](x,t^{n}) +\mathscr {O}(\tau ^2)\\= & {} UU_{x}(x,t^{n})+\tau \bigg [(\partial _{t}U)U_{x}+U(\partial _{t}U_{x})\bigg ](x,t^{n}) +\mathscr {O}(\tau ^2) \end{aligned}$$

Approximation of time derivative with finite differences accordingly the above equation gives us

$$\begin{aligned}{}[UU_{x}](x,t^{n}+\tau )= & {} UU_{x}(x,t^{n}) +\tau \bigg [\left( \frac{U^{n+1}-U^{n}}{\tau }+\mathscr {O}(\tau )\right) U^{n}_{x}\bigg ] \\&+\,\tau \bigg [\left( \frac{U_{x}^{n+1}-U_{x}^{n}}{\tau }+\mathscr {O}(\tau )\right) U^{n}\bigg ] +\mathscr {O}(\tau ^2) \end{aligned}$$

Simplification and re-arrangement then yield

$$\begin{aligned}{}[UU_{x}](x,t^{n}+\tau )= & {} UU_{x}(x,t^{n}) +\bigg [\left( U^{n+1}-U^{n}\right) U^{n}_{x}+\mathscr {O}(\tau ^2)\bigg ]\\&+\,\bigg [\left( U_{x}^{n+1}-U_{x}^{n}\right) U^{n}+\mathscr {O}(\tau ^2)\bigg ] +\mathscr {O}(\tau ^2)\\= & {} [U^{n}_{x}]U^{n+1}+[U^{n}]U_{x}^{n+1}-U^{n}U^{n}_{x}+\mathscr {O}(\tau ^2). \end{aligned}$$

Finally, omission of the $\mathscr {O}(\tau ^2)$ error term gives us the linearized Eq. (6). It is to mentioned that the same process has been used by Rashidinia and Rasoulizadeh [38] for linearization.