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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 this paper, we study the numerical integration method of Euler equation of high-order discontinuous finite element method on unstructured grids. First of all, according to the basic principle of the discontinuous finite element method, the numerical integral accuracy requirements of the neutral integral and the volume fraction of the discrete control equations of the finite element method are deduced. Then, the least integral number of points needed to satisfy the integral accuracy requirements is given when the linear integral term is processed by the Gauss-Legendre and Gauss-Lobatto integral formulas and the volume fraction is processed by the Guass integral formula and the reconstruction integral method . Finally, the requirements of the above integral accuracy are verified through concrete examples, and the effects of different numerical integration methods on the efficiency and accuracy of the solution are investigated.