High-order numerical method for the fractional Korteweg-de Vries equation using the discontinuous Galerkin method

The fractional Korteweg-de Vries (KdV) equation generalizes the classical KdV equation by incorporating truncation effects within bounded domains, offering a flexible framework for modeling complex phenomena. This paper develops a high-order, fully discrete local discontinuous Galerkin (LDG) method with generalized alternating numerical fluxes to solve the fractional KdV equation, enhancing applicability beyond the limitations of purely alternating fluxes. An efficient finite difference scheme approximates the fractional derivatives, followed by the LDG method for solving the equation. The scheme is proven unconditionally stable and convergent. Numerical experiments confirm the method's accuracy, efficiency, and robustness, highlighting its potential for broader applications in fractional differential equations.