基于二维非结构网格的GMRES隐式算法

GMRES Implicit Algorithm Based on 2D Unstructured Meshes for Solving Euler Equations

将广义极小残差GMRES(Generalized Minimum RESidual)隐式算法应用到二维非结构网格上,并结合LU-SGS(Lower Upper-Symmetric Gauss-Seidel)方法对所求解方程组的残值向量进行预处理,发展了一套高效、可靠的二维Euler方程的求解器。NACA0012翼型和某四段翼型的2个算例,表明该隐式算法的计算效率要比传统的四步Runge-Kutta显式算法高出几十倍,与LU-SGS隐式算法的效率相比,该算法的效率高出近1个量级。应用了重启型的GMRES算法,并对2种构造系数Jacobian矩阵的方法进行了比较。

Aim. Because of the linear property in the convergence speed of traditional explicit and some implicit schemes, computational efficiency based on unstructured meshes for complicated configuration is not satisfactory. Now we present a Generalized Minimum Residual （GMRES） implicit algorithm which has second order property in the convergence speed to solve Euler equations based on 2D unstructured meshes.The solution vector is obtained using Given＇s transform scheme with preconditioning the residual vector of equations using Lower-Upper Symmetric Gauss-Seidel （LU-SGS） method. Furthermore, local time stepping and implicit residual smoothing schemes are applied to develop an accurate, efficient and reliable solver. In the maximal eigenvalue splitting, Jacobian matrix is formulated firstly by variables of the center cell and its neighbor ceils, and secondly by variables on the public edges of them. The efficiency of the former method for formulating the Jacobian matrix is about a quarter higher than the latter. Compared with traditional four-stage Runge-Kutta explicit algorithm and LU-SGS implicit algorithm on test cases of NACA0012 airfoil and a 4-element airfoil, the results show that computational efficiency can be improved one or two magnitudes using GMRES-t-LU-SGS implicit algorithm. This algorithm can also be developed to 3D unstructured meshes to compute both viscous and inviscid flow.