高阶精度间断Galerkin有限元方法研究

Investigation of High-order Discontinuous Galerkin Finite Element Method

在二维非结构网格上,对高阶精度间断Galerkin有限元方法求解跨音速欧拉方程进行研究。运用间断有限元理论,采用施密特正交化多项式基函数对流场解进行近似描述,使用HLLC近似黎曼解方法计算网格单元边界处的数值通量,积分项通过高斯积分求解,时间推进采用经典四步Runge-Kutta方法,并引入斜率限制器,抑制数值振荡。对NACA0012翼型跨音速无粘流动进行数值模拟,数值结果表明:间断Galerkin有限元方法具有较高的精度,较小的数值耗散和较强的激波捕捉能力。

This paper deals with a high-order accurate discontinuous Galerkin finite element method for the numerical solution of the transonic Euler equations.Based on the theory of discontinuous finite element method,within each element the solutions are expanded in a series of Schmidt orthogonal polynomials base functions.The nonlinear numerical flux is discretized by using the HLLC fluxs and the integration terms are calculated by using Gaussian quadrature rules;Time is advanced by explicit forth-order accurate Runge-Kutta method and slope limiter is constructed to suppress numerical oscillations.Transonic inviscid flow over a NACA0012 airfoil was simulated by this method.The numerical results indicate that the discontinuous Galerkin method has properties of high-order accuracy,smaller numerical dissipation and excellent ability to capture shocks.