GMRES implicit algorithm based on 2D unstructured meshes for solving Euler equations

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 cells, 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+LU-SGS implicit algorithm. This algorithm can also be developed to 3D unstructured meshes to compute both viscous and inviscid flow.