Equivalence Between DFR Method and LDG Method for Solving Linear Third-Order KdV Equation

The equivalence between direct flux reconstruction method and local discontinuous Galerkin method in solving linear third-order KdV equation is studied. Firstly, the linear third-order KdV equation is spatially dispersed by DFR method and LDG method respectively, and the spatial discretization schemes of two numerical algorithms are given. Secondly, the equivalence of DFR and LDG methods is proved by two methods. The first proof relies on the property of Gauss quadrature: the M-point Gauss quadrature has 2M - 1 order algebraic accuracy; the second proof takes advantage of the special property of the Lobatto polynomial : the zeros of the derivative of Lobatto polynomial of degree M + 1 are M Gauss points. At last, taking the value of M to be 1 as an example, it is shown that the two numerical algorithms are equivalent to the ordinary differential equations used for programming when solving the linear third-order KdV equation.