基于Newton/Gauss-Seidel迭代的DGM隐式方法

在Newton迭代方法的基础上,对高阶精度间断Galerkin有限元方法(DGM)的时间隐式格式进行了研究. Newton迭代 法的优势在于收敛效率高效,并且定常和非定常问题能够统一处理,对于非定常问题无需引入双时间步策略. 为了避免大型矩阵的求逆,采用一步Gauss-Seidel迭代和Matrix-free技术消去残值Jacobi矩阵的上、下三角矩阵,从而只需计算和存储对角(块)矩阵. 对角(块)矩阵采用数值方法计算. 空间离散采用Taylor基,其优势在于对于任意形状的网格,基函数的形式是一致的,有利于在混合网格上推广. 利用该方法,数值模拟了Bump绕流和NACA0012翼型绕流. 计算结果表明,与显式的Runge-Kutta时间格式相比,隐式格式所需的迭代步数和CPU时间均在很大程度上得到减少,计算效率能够提高1～ 2个量级.

AN IMPLICIT ALGORITHM FOR DISCONTINUOUS GALERKIN METHOD BASED ON NEWTON/GAUSS-SEIDEL ITERATIONS

An effcient implicit algorithm was developed for high-order discontinuous Galerkin(DGM) based on Newton/Gauss-Seidel iteration approach.The second-order to the forth order DGMs based on Taylor basis functions were employed to carry out the spatial discretization.Newton iteration scheme was used to solve the nonlinear system,and the linear system was solved with one-step Gauss-Seidel iteration.In addition,the effects of several parameters in the implicit scheme,such as the CFL number,the Newton sub-iteration steps,and the update frequency of mass-matrix,have been investigated for two-dimensional Euler equations.Two typical cases,including subsonic flows over a bump and a NACA0012 airfoil,were simulated,and compared with the traditional explicit Runge-Kutta scheme.The numerical results demonstrate that the present implicit scheme can accelerate the convergence history evidently.