本文主要针对高阶偏微分方程，构造高精度数值方法并进行理论分析。 文章分为两个方面，一是构造整体高精度数值格式，对全离散格式做误差分析。二是对谱体积方法的半离散格式做超收敛分析。 首先，我们对一维Korteweg-de Vries (KdV)方程设计哈密尔顿量守恒的整体高阶精度数值格式，其中时间离散使用time-stepping谱方法，空间采用局部间断有限元(LDG)方法。谱方法优势在于其高阶谱精度，且相比于h-version的数值格式调整网格尺寸，这类p-version的格式更注重多项式的次数，在长时间模拟方面其计算效率优势尤为明显。我们设计了两种时间谱方法，即，Spectral Petrov-Galerkin方法(SPG)和Gauss配点法(SGC)。针对广义KdV方程，全离散SPG-LDG数值格式可以保持动量和哈密尔顿量守恒。而全离散SGC-LDG方法对于线性KdV方程可以保持动量和哈密尔顿量守恒，对于非线性KdV方程，保持动量守恒，哈密尔顿量近似守恒，能量误差可以达到谱精度。全离散SGC-LDG和SPG-LDG方法在空间和时间都具有谱精度，即误差都以多项式次数指数衰减。数值实验结果进一步验证了我们的理论结果，SPG-LDG和SGC-LDG方法均能保持动量，L2能量和哈密尔顿量守恒，并且长时间模拟中，图形相位和形状不发生改变。 其次，针对一维和二维扩散方程的半离散谱体积(SV)方法进行了超收敛分析。算法的核心思想在于引入辅助变量，将方程改写为等价的一阶系统，之后使用SV方法求解系统。基于控制体选取的不同，设计了两种形式的SV方法，一种是基于Gauss点的GSV方法，一种是基于Radau 点的RSV方法。通过构造修正函数以及合适的初值离散，我们严格地证明了对于一维问题，特殊插值函数与GSV数值解的误差在L2范数意义下有2k阶超收敛， RSV数值解有2k+1阶超收敛，其中k是多项式的最高项次数。借助此项超收敛结果，我们进一步证明了SV方法数值解在区间平均值，以及流通量在节点的误差以2k+1 (2k)阶速度超收敛。此外，还发现了数值解和其导数在Gauss点或者Radau点的超收敛性质。对于二维问题，证明了特殊插值函数与GSV和RSV数值解的误差在L2范数意义下有k+2阶超收敛，数值算例验证了我们的理论结果。

    In this thesis, the main work includes constructing high accuracy numerical methods and theoretical analyses for higher order partial  differential equations.  This work is mainly divided into two aspects. One is to construct the  high accuracy numerical scheme and analyze the error of the fully-discrete scheme. The other is the superconvergence analysis of the semi-discrete scheme of spectral volume method.

  The first part is that  we design a global high order accuracy numerical scheme with Hamiltonian energy preserving for one-dimensional Korteweg-de Vries equations, in which time-stepping spectral Petrov-Galerkin (SPG) or Gauss collocation (SGC) methods for the temporal discretization coupled with the p-version/spectral local discontinuous Galerkin (LDG) methods for the space discretization. The advantage of time-stepping spectral  method lies in its high order spectral accuracy. Compared with the numerical scheme of h-version to adjust the mesh size, this p-version scheme pays more attention to the degree of polynomial, and its computational efficiency advantage is particularly significant in long time calculation simulation.We prove that the fully-discrete SPG-LDG scheme preserves both the momentum and the Hamiltonian energy exactly for generalized KdV equations. While the fully-discrete SGC-LDG formulation preserves the momentum and the Hamiltonian energy exactly for linearized KdV equations. As for nonlinear KdV equations, the SGC-LDG scheme preserves the momentum exactly and is Hamiltonian conserving up to some spectral accuracy. Furthermore, we show that the fully-discrete SGC-LDG and SPG-LDG methods converge exponentially with respect to the polynomial degree in both space and time for linear KdV equations. The numerical experiments are provided to demonstrate that the proposed numerical methods preserve the momentum, L2 energy and Hamiltonian energy and maintain the shape of the solution phase efficiently over long time period.
     The second part is that we perform the superconvergence analysis of the semi-discrete spectral volume method (SV) for one-dimensional and two-dimensional diffusion equation.
 The main idea of the numerical scheme is to rewrite the diffusion equation into an equivalent first-order system, and then use the
 SV method to solve the system. Superconvergence property of two SV schemes based on alternating fluxes are investigated, which are  designed  by using the Gauss points or  Radau points of the underlying meshes to  construct control volumes.  For one-dimensional diffusion equation, by constructing  correction functions and suitable initial value discretizations, we strictly prove that  the errors between the  special interpolation and GSV numerical solutions  have  2k th superconvergence rate in the sense of L2 norm, and RSV numerical solutions have  2k+1 th superconvergence rate, when piecewise polynomials of degree k are used. The 2k + 1 th (or 2k th) superconvergence rate for the error of numerical fluxes at nodes  and for the cell average error  are obtained. Furthermore,  interior superconvervgence points for both the function value and derivative value approximations
 are discovered, which are identified as Gauss points or Radau points.  For two-dimensional diffusion equation, the k + 2 th  superconvergence rate for the error between numerical solution  and the special interpolation   is proved. Numerical experiments are presented to validate our  theoretical findings.