A Schur complement formulation for solving free-boundary,Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm–Newton approach, which employs Newton’s method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt–solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.     

PROGRAM-PATTERN MULTIPOLE BOUNDARY ELEMENT METHOD FOR FRICTIONAL CONTACT

A mathematical program is proposed for the highly nonlinear problem involving frictional contact. A program-pattern using the fast multipole boundary element method (FMBEM) is given for 3-D elastic contact with friction to replace the Monte Carlo method. A newoptimized generalized minimal residual (GMRES) algorithm is presented. Numerical examples demonstrate the validity of the program-pattern optimization model for node-to-surface contact with friction. The GMRES algorithm greatly improves the computational efficiency.     

Any admissible cycle‐convergence behavior is possible for restarted GMRES at its initial cycles

We show that any admissible cycle‐convergence behavior is possible for restarted GMRES at a number of initial cycles, moreover the spectrum of the coefficient matrix alone does not determine this cycle‐convergence. The latter can be viewed as an extension of the result of Greenbaum, Pták and Strako? (SIAM Journal on Matrix Analysis and Applications 1996; 17 (3):465–469) to the case of restarted GMRES. Copyright © 2010 John Wiley & Sons, Ltd.     

跳跃扩散下双障碍期权定价的数值解

提出了一种求解带有跳跃的双障碍期权定价模型的数值方法.算法采用了Crank-Nicolson 有限差分格式和复化梯形公式对模型进行离散,对离散后的线性系统采用GMRES迭代法求解,并且构造了一个新的预处理算子以加速迭代法的收敛.数值实验验证了该方法能快速求解模型并达到二阶收敛精度.     

On IGMRES: An incomplete generalized minimal residual method for large unsymmetric linear systems

The truncated version of the generalized minimal residual method (GMRES), the incomplete generalized minimal residual method (IGMRES), is studied. It is based on an incomplete orthogonalization of the Krylov vectors in question, and gives an approximate or quasi-minimum residual solution over the Krylov subspace. A convergence analysis of this method is given, showing that in the non-restarted version IGMRES can behave like GMRES once the basis vectors of Krylov subspace generated by the incomplete orthogonalization are strongly linearly independent. Meanwhile, some relationships between the residual norms for IOM and IGMRES are established. Numerical experiments are reported to show convergence behavior of IGMRES and of its restarted version IGMRES(m). Project supported by the China State Key Basic Researches, the National Natural Science Foundation of China (Grant No. 19571014), the Doctoral Program (97014113), the Foundation of Returning Scholars of China and the Natural Science Foundation of Liaoning Province.     

Uniformly Convergent Multigrid Methods for Convection–Diffusion Problems without Any Constraint on Coarse Grids

We construct a class of multigrid methods for convection–diffusion problems. The proposed algorithms use first order stable monotone schemes to precondition the second order standard Galerkin finite element discretization. To speed up the solution process of the lower order schemes, cross-wind-block reordering of the unknowns is applied. A V-cycle iteration, based on these algorithms, is then used as a preconditioner in GMRES. The numerical examples show that this method is convergent without imposing any constraint on the coarsest grid and the convergence of the preconditioned method is uniform.     

一种新型压缩预处理CGS算法

CGS算法是求解大型非对称线性方程组的常用算法，然而该算法无极小残差性质，因此它常因出现较大的中间剩余向量而出现典型的不规则收敛行为．本根据IRA方法提出了一种压缩预处理CGS方法，数值实验表明这种算法在一定程度上减小了迭代算法在收敛过程中的剩余问题，从而使得算法具有更好的稳定性，该法构造简单，减少了收敛次数，加快了收敛速度．     

Sine transform based preconditioning techniques for space fractional diffusion equations

We study the preconditioned iterative methods for the linear systems arising from the numerical solution of the multi-dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed according to the symmetric and skew-symmetric splitting of the Toeplitz factor in the resulting coefficient matrix. Theoretical analyses show that the upper bound of relative residual norm of the GMRES method when applied to the preconditioned linear system is mesh-independent which implies the linear convergence. Numerical experiments are carried out to illustrate the correctness of the theoretical results and the effectiveness of the proposed preconditioning technique.     

预条件GMRES(m)法迭代求解大规模边界元弹性问题

提出了一种基于单元节点的块雅可比预条件方法,扩大了边界元法的计算规模,使之可用于大规模工程问题的求解.数值实验说明了这种预条件技术的有效性,表明预条件GMRES(m)算法具有较好的收敛特性,适合于求解大规模问题边界元弹性问题所形成的稠密非对称线性方程组.     

A 3‐D non‐hydrostatic pressure model for small amplitude free surface flows

A three‐dimensional, non‐hydrostatic pressure, numerical model with k–ε equations for small amplitude free surface flows is presented. By decomposing the pressure into hydrostatic and non‐hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step the momentum equations are solved without the non‐hydrostatic pressure term, using Newton's method in conjunction with the generalized minimal residual (GMRES) method so that most terms can be solved implicitly. This method only needs the product of a Jacobian matrix and a vector rather than the Jacobian matrix itself, limiting the amount of storage and significantly decreasing the overall computational time required. In the second step the pressure–Poisson equation is solved iteratively with a preconditioned linear GMRES method. It is shown that preconditioning reduces the central processing unit (CPU) time dramatically. In order to prevent pressure oscillations which may arise in collocated grid arrangements, transformed velocities are defined at cell faces by interpolating velocities at grid nodes. After the new pressure field is obtained, the intermediate velocities, which are calculated from the previous fractional step, are updated. The newly developed model is verified against analytical solutions, published results, and experimental data, with excellent agreement. Copyright © 2005 John Wiley & Sons, Ltd.