矩阵带宽缩减技术在隐式间断有限元中的应用

为了数值求解二维Euler方程，以间断有限元方法作为空间离散、向后差分公式（BDF）作为时间离散。针对采用牛顿法求解源于隐式时间积分的非线性方程组，构造了相应的Jacobi矩阵，其具有阶数高、稀疏性强、数值非对称的特点。在每个时间步内，选择带预处理的广义极小残量（GMRES）方法求解线性方程组，预处理矩阵由不完全LU分解（ILU）方法构造。将矩阵带宽缩减技术应用于上述求解过程，无需额外的存储空间，就缩小了预处理矩阵与系数矩阵的差距，从而加快了GMRES方法的收敛、增大了可用的时间步长。通过求解典型的空气动力学问题，检验了该应用的有效性。 

Application of matrix bandwidth reduction technique in implicit discontinuous Galerkin

To numerically solve the two-dimensional Euler equations, discontinuous Galerkin method and backward difference formula (BDF) are used as spatial and temporal discretization, respectively. The Newton-Raphson method is taken to solve the nonlinear equations arising from the implicit time integration. The Jacobia matrix is constructed. Owing to the high-order, sparsity and non-symmetry of the matrix, the preconditioned generalized minimal residual (GMRES) method is chosen in every time step for solving the linear equations. The preconditioner is constructed using incomplete lower-upper (ILU) decomposition method. The bandwidth reduction technique is applied to the solution of the linear equations. Without extra storage cost, the application narrows the difference between the preconditioner and the coefficient matrix, thus accelerating the convergence of GMRES method and increasing the available time step size for temporal integration. Typical aerodynamic problems are solved to test the effectiveness of the application.