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.

在本文中，我们分析了半离散超弱局部不连续伽辽金（UWLDG）方法对一维瞬态线性五阶方程的超收敛性。UWLDG 方法旨在求解具有高阶空间导数的方程。主要思想是将高阶方程改写为低阶系统。当我们使用UWLDG方法求解五阶方程时，我们将其改写为具有两个二阶方程和一个一阶方程的系统。与其他关于 DG 方法超收敛的工作相比，主要挑战是为包含不同阶方程的系统定义校正函数和特殊插值函数。我们根据\(k\pmod {5}\)将我们的分析分为五种情况，其中k是我们函数空间中多项式的最高次数，并获得单元边界处单元平均值和函数值的2 k阶超收敛，以及某些特殊正交点处函数值的\(k+2\)阶超收敛。对于两个二阶方程的数值解，我们证明一阶导数在单元边界处具有2 k阶超收敛性，在一类特殊正交点处具有阶数\(k+1 \)。所有的理论结果都得到了数值实验的证实。