一维激波管问题的无数值积分间断伽辽金法数值模拟

Numerical Simulation of Shock Tube Problem Using Quadrature-free Discontinuous Galerkin Method

本文利用间断伽辽金法对一维激波管问题进行了数值模拟。在使用间断伽辽金法过程中,会引入额外的积分项,对于这些积分项,每一时间步进行数值积分需要大量的计算时间,为了克服这个问题并优化相应的求解器,提出了一种基于Lagrange插值多项式的无数值积分格式的间断伽辽金法,这样所需计算时间以及内存将显著减少,同时应用也变得更为简便。最后,利用无数值积分间断伽辽金法,对算例进行了数值模拟并与精确解对比,取得很好的吻合效果。

In this paper, discontinuous Galerkin method （DGM） is used to numerically simulate onedimensional shock tube problems. It is noted that implementation of DGM will produce extra integral term, which gives rise to additional computation overhead because of the use of numerical quadrature in every time step. To overcome this problem and optimize the DG solver, we develop a new DGM with quadrature-free formulation based upon Lagrange interpolate polynomials. Using this new scheme, the required computational time and storage memory can be significantly reduced and implementation can be simpler. Finally, the numerical results are made and compared with exact solutions, satisfactory results are obtained.