分数阶Korteweg-de Vries方程零色散极限的谱Galerkin方法
Spectral Galerkin method for the zero dispersion limit of the fractional Korteweg-de Vries equation
AI总结：
本文提出一种守恒且稳定的全离散Crank-Nicolson Fourier-谱-Galerkin格式求解分数阶KdV方程，证明其收敛性并验证零色散极限收敛到Hopf方程解，数值实验支持理论结果。
摘要：
我们提出了一种全离散的Crank-Nicolson Fourier-谱-Galerkin (FSG)格式，用于逼近分数阶Korteweg-de Vries (KdV)方程的解，该方程涉及指数为$\alpha\in [1,2]$的分数阶Laplacian和阶数为$\varepsilon^2$的小色散系数。当$\varepsilon \to 0$时的极限解被称为零色散极限。我们证明了半离散FSG格式守恒前三个积分不变量，从而保持结构，并且全离散FSG格式是$L^2$-守恒的，确保稳定性。利用紧性论证，我们构造性地证明了近似解收敛到分数阶KdV方程在$C([0,T]; H_p^{1+\alpha}(\mathbb{R}))$中的唯一解，对于$H_p^{1+\alpha}(\mathbb{R})$中的周期初值。所设计的格式对于$H_p^r$（$r \geq 1+\alpha$）中的初值达到谱精度，对于解析初值达到指数精度。此外，我们建立了由全离散FSG格式获得的零色散极限的近似在$\varepsilon \to 0$时在$L^2$中收敛到Hopf方程的解，直到梯度灾变时间$t_c$。在$t_c$之后，数值研究表明近似收敛到渐近解，该解在$\alpha= 2$的振荡区域内由Whitham平均方程弱描述。提供了数值结果以证明该格式的收敛性并验证理论发现。
英文摘要：
We present a fully discrete Crank-Nicolson Fourier-spectral-Galerkin (FSG) scheme for approximating solutions of the fractional Korteweg-de Vries (KdV) equation, which involves a fractional Laplacian with exponent $α\in [1,2]$ and a small dispersion coefficient of order $\varepsilon^2$. The solution in the limit as $\varepsilon \to 0$ is known as the zero dispersion limit. We demonstrate that the semi-discrete FSG scheme conserves the first three integral invariants, thereby structure preserving, and that the fully discrete FSG scheme is $L^2$-conservative, ensuring stability. Using a compactness argument, we constructively prove the convergence of the approximate solution to the unique solution of the fractional KdV equation in $C([0,T]; H_p^{1+α}(\mathbb{R}))$ for the periodic initial data in $H_p^{1+α}(\mathbb{R})$. The devised scheme achieves spectral accuracy for the initial data in $H_p^r,$ $r \geq 1+α$ and exponential accuracy for the analytic initial data. Additionally, we establish that the approximation of the zero dispersion limit obtained from the fully discrete FSG scheme converges to the solution of the Hopf equation in $L^2$ as $\varepsilon \to 0$, up to the gradient catastrophe time $t_c$. Beyond $t_c$, numerical investigations reveal that the approximation converges to the asymptotic solution, which is weakly described by the Whitham's averaged equation within the oscillatory zone for $α= 2$. Numerical results are provided to demonstrate the convergence of the scheme and to validate the theoretical findings.
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