高阶间断有限元方法求解Euler方程时的数值积分问题研究

Integration Strategy for High-Order Discontinuous Galerkin(DG) Method for Solving Euler Equations

文章对非结构网格上高阶间断有限元方法求解Euler方程时的数值积分方式进行了研究。首先根据间断有限元方法的基本原理推导出了有限元离散控制方程中线积分和体积分项的数值积分精度要求。然后给出了采用Gauss-Legendre和Gauss-Lobatto积分公式处理线积分项,以及采用Guass积分公式和重构积分方法处理体积分项的情况下,为满足积分精度要求所需使用的最少积分节点数目。最后,通过具体算例对上述积分精度要求进行了验证,并考察了不同数值积分方法对于求解效率和精度的影响。

In the full paper, according to the DG method, we derive the accuracy requirements of numerical integration for line integral and volume integral in the Euler equations discretized by unstructured grid. Considering the above accuracy requirements, we summarize ： （ 1 ） the minimal numbers of integral nodes used in the line integral with the Gauss-Legendre quadrature rule and the Gauss-Lobatto quadrature rule respectively, （2） the minimal number of integral nodes used in the volume integral with the Gauss quadrature rule, （3） a new quadrature method based on reconstruction. By using a numerical example, we evaluate： （1） the accuracy requirements for the numerical integration method, （2） the effects of different numerical integrations on computing stability, accuracy and efficiency. The numerical simulation results, given in Figs. 2 through 6, and their analysis show preliminarily that ： （ 1 ） in case the accuracy requirement of line integral is not fulfilled, the stability and the convergence of DG meth- od can not be ensured ; the accuracy requirement of volume integral has strong influence on the accuracy of flow solution; （2） the Gauss-Legendre quadrature rule is more efficient than the Gauss-Lobatto quadrature rule for the same accuracy of numerical integration; （3） for high-order DG discretization, our quadrature method based on re- construction is more accurate than the classical Gauss quadrature rule although its efficiency is lower by approxi- mately 15%.