In this paper we investigate three various algorithms for computation of generalized inverses which are contained in the limit expressions $\mathop{\rm lim}\limits_{z\to0} V (DT + zI)^{-l} U$ and $\mathop{\rm lim}\limits_{z\to0} V (DT + zI)^{-l} Z^e.$ These algorithms are extensions of the algorithms developed by various authors in [2], [3-4], [7-9], [16-18].

This paper continues to study the central relaxing schemes for system of hyperbolic conservation laws, based on the local relaxation approximation. Two classes of relaxing systems with stiff source term are introduced to approximate system of conservation laws in curvilinear coordinates. Based on them, the semi-implicit relaxing schemes are constructed as in [6,12] without using any linear or nonlinear Riemann solvers. Numerical experiments for one-dimensional and two-dimensional problems are presented to demonstrate the performance and resolution of the current schemes.  

This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise $C^1$ problems and splitting methods can be explained uniformly with the same viewpoint.  

This paper suggests a probabilistic numerical approach for a class of PDE. First of all, by simulating Brownian motion and using Monte-Carlo method, we obtain a probabilistic numerical solution for the PDE. Then, we prove that the probabilistic numerical solution converges in probability to its solution. At the end of this paper, as an application, we give a probabilistic numerical approach for the valuation of European Options, where we see volatility $\sigma$, interest rate $r$ and divident rate $D_0$ as functions of stock $S$, respectively. 

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the error analysis of one-leg methods applied nonlinear stiff DDEs. It is proved that an A-stable one-leg method with a simple linear interpolation is D-convergent of order $p$, if it is consistent of order $p$ in the classical sense.

A global finite element nonlinear Galerkin method for the penalized Navier-Stokes equations is presented. This method is based on two finite element spaces $X_H$ and $X_h$, defined respectively on one coarse grid with grid size $H$ and one fine grid with grid size $h＜＜H$. Comparison is also made with the finite element Galerkin method. If we choose $H=O(ε^{-1/4}h^{1/2}), ε＞0$ being the penalty parameter, then two methods are of the same order of approximation. However, the global finite element nonlinear Galerkin method is much cheaper than the standard finite element Galerkin method. In fact, in the finite element Galerkin method the nonlinearity is treated on the fine grid finite element space $X_h$ and while in the global finite element nonlinear Galerkin method the similar nonlinearity is treated on the coarse grid finite element space $X_H$ and only the linearity needs to be treated on the fine grid increment finite element space $W_h$. Finally, we provide numerical test which shows above results stated.

This paper is devoted to computation of hypersonic flow of air with chemical reactions over concave corners. A technique combining smooth transformation of domain and implicit difference methods is used to overcome numerical difficulties associated with the lack of resolution behind the shock and near the body. The implicit treatment of right hand side terms is also an important part of our method.

In this paper, we construct the genuine-optimal circulant preconditioner for finite-section Wiener-Hopf equations. The genuine-optimal circulant preconditioner is defined as the minimizer of Hilbert-Schmidt norm over certain integral operators. We prove that the difference between the genuine-optimal circulant preconditioner and the original integral operator is the sum of a small norm operator and a finite rank operator. Thus, the preconditioned conjugate gradient (PCG) method, when applied to solve the preconditioned equations, converges superlinearly. Finally, we give an efficient algorithm for the solution of Wiener-Hopf equation discretized by high order quadrature rules.  

This paper deals with the solution of partitioned systems of nonlinear stiff differential equations. Given a differential system, the user may specify some equations to be stiff and others to be nonstiff. For the numerical solution of such a system Parallel Compound Methods (PCMs) are studied. Nonstiff equations are integrated by a parallel explicit RK method while a parallel Rosenbrock method is used for the stiff part of the system.
Their order conditions, their convergence and their numerical stability are discussed, and the numerical tests are conducted on a personal computer and a parallel computer.  

For the large sparse system of linear equations with symmetric positive definite block coefficient matrix resulted from suitable finite element discretization of the second-order self-adjoint elliptic boundary value problem, by making use of the algebraic multilevel iteration technique and the blocked preconditioning strategy, we construct preconditioning matrices having parallel computing function for the coefficient matrix and set up a class of parallel hybrid algebraic multilevel iteration methods for solving this kind of system of linear equations. Theoretical analyses show that, besides much suitable for implementing on the high-speed parallel multiprocessor systems, these new methods are optimal-order methods. That is to say, their convergence rates are independent of both the sizes and the levels of the constructed matrix sequence, and the computational workloads are bounded by linear functions in the order number of the considered system of linear equations, respectively.