Spectral Galerkin method for the zero dispersion limit of the fractional Korteweg-de Vries equation
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 $\alpha \in [1,2]$ 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 $C([0,T]; H_p^{1+\alpha}(\mathbb{R}))$ for the periodic initial data in $H_p^{1+\alpha}(\mathbb{R})$. The devised scheme achieves spectral accuracy for the initial data in $H_p^r,$ $r \geq 1+\alpha$ 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 $\alpha = 2$. Numerical results are provided to demonstrate the convergence of the scheme and to validate the theoretical findings.