Numerical Analysis of the Fractional Seventh-Order KdV Equation Using an Implicit Fully Discrete Local Discontinuous Galerkin Method

Abstract

In this paper an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is applied to solve the time-fractional seventh-order Korteweg-de Vries (sKdV) equation, which is introduced by replacing the integer-order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(\Delta t)^2+(\Delta t)^{\frac{\alpha}{2}}h^{k+\frac{1}{2}})$ through analysis. Extensive numerical results are provided to demonstrate the performance of the present method.