Superconvergence Analysis of the Ultra-Weak Local Discontinuous Galerkin Method for One Dimensional Linear Fifth Order Equations

In this paper, we analyze the superconvergence of the semi-discrete ultra-weak local discontinuous Galerkin (UWLDG) method for one dimensional time-dependent linear fifth order equations. The UWLDG method is designed to solve equations with high order spatial derivatives. The main idea is to rewrite the higher order equation into a lower order system. When we use the UWLDG method to solve the fifth order equations, we rewrite it as a system with two second order equations and one first order equation. Compared with the other works about superconvergence of the DG method, the main challenge is to define correction functions and a special interpolation function for the system containing equations with different orders. We divide our analysis into five cases according to $$k\pmod {5}$$ , where k is the highest degree of polynomials in our function space, and obtain 2k-th order superconvergence for cell averages and function values at the cell boundaries and $$k+2$$ -th order superconvergence for function values at some special quadrature points. For numerical solutions of the two second order equations, we prove that the first derivatives have superconvergence of order 2k at cell boundaries and order $$k+1 $$ at a class of special quadrature points. All theoretical results are confirmed by numerical experiments.