On stable implicit difference scheme for hyperbolic–parabolic equations in a Hilbert space

The first‐order of accuracy difference scheme for approximately solving the multipoint nonlocal boundary value problem for the differential equation in a Hilbert space H, with self‐adjoint positive definite operator A is presented. The stability estimates for the solution of this difference scheme are established. In applications, the stability estimates for the solution of difference schemes of the mixed type boundary value problems for hyperbolic–parabolic equations are obtained. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the positive definite solutions of nonlinear matrix equation X + A⋆ X−δA = Q

In this paper we consider the positive definite solutions of nonlinear matrix equation X + A^⋆X^−δA = Q, where δ ∈ (0, 1], which appears for the first time in [S.M. El-Sayed, A.C.M. Ran, On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions of the equation is discussed. The error estimations are found.     

A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Auⁿ⁺¹ = uⁿ, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au¹ = u⁰, contains the solutions {uⁿ} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.     

Positive fixed points for a class of nonlinear operators and applications

In this paper, we study the existence and uniqueness of positive solutions for a class of nonlinear operator equations on ordered Banach spaces. Various applications are also considered to illustrate our obtained results (existence of solutions to quadratic integral equations with a linear modification of the argument, positive solution of second-order Neumann boundary value problem, and positive definite solutions of a class of nonlinear matrix equations).     

Two-grid stabilized algorithms for the steady Navier–Stokes equations with damping

This paper presents and studies three two-grid stabilized quadratic equal-order finite element algorithms based on two local Gauss integrations for the steady Navier–Stokes equations with damping. In these algorithms, we first solve a stabilized nonlinear problem on a coarse grid, and then pass the coarse grid solution to a fine grid and solve a stabilized linear problem. Using some nonlinear analysis techniques, we analyze stability of the algorithms and derive optimal order error estimates of the approximate solutions. Theoretical and numerical results show that, when the algorithmic parameters are chosen appropriately, the accuracy of the approximate solutions computed by our two-grid stabilized algorithms is comparable to that of solving a fully stabilized nonlinear problem on the same fine grid; however, our two-grid algorithms save a large amount of CPU time than the one-grid stabilized algorithm.     

A second‐order finite difference approximation for a mathematical model of erythropoiesis

We present a second‐order finite difference scheme for approximating solutions of a mathematical model of erythropoiesis, which consists of two nonlinear partial differential equations and one nonlinear ordinary differential equation. We show that the scheme achieves second‐order accuracy for smooth solutions. We compare this scheme to a previously developed first‐order method and show that the first order method requires significantly more computational time to provide solutions with similar accuracy. We also compare this numerical scheme with other well‐known second‐order methods and show that it has better capability in approximating discontinuous solutions. Finally, we present an application to recovery after blood loss. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析

考虑非线性矩阵方程X A~*X~(-n)A=P,其中A是m阶非奇异复矩阵,P是m阶Hermite正定矩阵.本文利用不动点理论讨论了该方程Hermite正定解的存在性及包含区间,给出了极大解的性质及求极大,极小解的迭代算法.研究了极大解的扰动问题,利用微分等方法获得了两个新的一阶扰动界,并给出数值例子对所得结果进行了比较说明.     

Computable error bounds and estimates for the conjugate gradient method

The conjugate gradient method is one of the most popular iterative methods for computing approximate solutions of linear systems of equations with a symmetric positive definite matrix A. It is generally desirable to terminate the iterations as soon as a sufficiently accurate approximate solution has been computed. This paper discusses known and new methods for computing bounds or estimates of the A-norm of the error in the approximate solutions generated by the conjugate gradient method.     

Two remarks on Monge-Ampere equations

Summary We consider real Monge-Ampère equations and we present two new properties of these equations. First, we show the existence of the «first eigenvalue of Monge-Ampère equation» i.e. we show the existence of a positive constant possessing all the properties of the first eigenvalue of a 2-nd order elliptic operator (positivity, uniqueness of the eigenfunction, maximum principle, bifurcation...).The second property concerns variational characterisations of solutions. Both properties are closely related to similar properties of the general class of Hamilton-Jacobi-Bellman equations.     

An efficient method for solving non-linear singularly perturbed two points boundary-value problems of fractional order

In this paper, we discuss a numerical solution of a class of non-linear fractional singularly perturbed two points boundary-value problem. The method of solution consists of solving reduced problem and boundary layer correction problem. A series method is used to solve the boundary layer correction problem, and then the series solutions is approximated by the Pade’ approximant of order [m, m]. Some theoretical results are established and proved. Two numerical examples are discussed to illustrate the efficiency of the present scheme.     

Preserving monotonicity in the numerical solution of Riccati differential equations

Summary. Solutions of symmetric Riccati differential equations (RDEs for short) are in the usual applications positive semidefinite matrices. Moreover, in the class of semidefinite matrices, solutions of different RDEs are also monotone, with respect to properly ordered data. Positivity and monotonicity are essential properties of RDEs. In Dieci and Eirola (1994), we showed that, generally, a direct discretization of the RDE cannot maintain positivity, and be of order greater than one. To get higher order, and to maintain positivity, we are thus forced to look into indirect solution procedures. Here, we consider the problem of how to maintain monotonicity in the numerical solutions of RDEs. Naturally, to obtain order greater than one, we are again forced to look into indirect solution procedures. Still, the restrictions imposed by monotonicity are more stringent that those of positivity, and not all of the successful indirect solution procedures of Dieci and Eirola (1994) maintain monotonicity. We prove that by using symplectic Runge-Kutta (RK) schemes with positive weights (e.g., Gauss schemes) on the underlying Hamiltonian matrix, we eventually maintain monotonicity in the computed solutions of RDEs. Received May 2, 1995     

Some iterative algorithms for positive definite solution to nonlinear matrix equations

This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms.     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

Energy-conserved splitting FDTD methods for Maxwell’s equations

In this paper, two new energy-conserved splitting methods (EC-S-FDTDI and EC-S-FDTDII) for Maxwell’s equations in two dimensions are proposed. Both algorithms are energy-conserved, unconditionally stable and can be computed efficiently. The convergence results are analyzed based on the energy method, which show that the EC-S-FDTDI scheme is of first order in time and of second order in space, and the EC-S-FDTDII scheme is of second order both in time and space. We also obtain two identities of the discrete divergence of electric fields for these two schemes. For the EC-S-FDTDII scheme, we prove that the discrete divergence is of first order to approximate the exact divergence condition. Numerical dispersion analysis shows that these two schemes are non-dissipative. Numerical experiments confirm well the theoretical analysis results.     

Reducible quadrature methods for Volterra integral equations of the first kind

Quadrature rules, generated by linear multistep methods for ordinary differential equations, are employed to construct a wide class of direct quadrature methods for the numerical solution of first kind Volterra integral equations. Our class covers several methods previously considered in the literature. The methods are convergent provided that both the first and second characteristic polynomial of the linear multistep method satisfy the root condition. Furthermore, the stability behaviour for fixed positive values of the stepsizeh is analyzed, and it turns out that convergence implies (fixedh) stability. The subclass formed by the backward differentiation methods up to order six is discussed and illustrated with numerical examples.     

Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations

Based on two-grid discretizations, some local and parallel finite element algorithms for the d-dimensional (d = 2,3) transient Stokes equations are proposed and analyzed. Both semi- and fully discrete schemes are considered. With backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Stokes equations using a coarse grid on the entire domain, then correct the resulted residue using a finer grid on overlapped subdomains by some local and parallel procedures at each time step. By the technical tool of local a priori estimate for the fully discrete finite element solution, errors of the corresponding solutions from these algorithms are estimated. Some numerical results are also given which show that the algorithms are highly efficient.     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations

We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off‐diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.     

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.     

Divergence-free wavelet solution to the Stokes problem

In this paper, we use divergence-free wavelets to give an adaptive solution to the velocity field of the Stokes problem. We first use divergence-free wavelets to discretize the divergence-free weak formulation of the Stokes problem and obtain a discrete positive definite linear system of equations whose coefficient matrix is quasi-sparse; Secondly, an adaptive scheme is used to solve the discrete linear system of equations and the error estimation and complexity analysis are given.