For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretica...For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretical analysis shows that the ARHSS method converges unconditionally to the unique solution of the saddle point problem. Finally, we use a numerical example to confirm the effectiveness of the method.展开更多
In this paper,we consider the so-called "inexact Uzawa" algorithm applied to the unstable Navier-Stokes problem.We use stabilization matrix to stabilize the unstable system and proved theoretically that unde...In this paper,we consider the so-called "inexact Uzawa" algorithm applied to the unstable Navier-Stokes problem.We use stabilization matrix to stabilize the unstable system and proved theoretically that under given proper preconditioners,Uzawa algorithm is convergent for the stablization system.Bounds for the iteration error are provided.We show numerically that Uzawa algorithm is convergent as well for the stabilization systems when it is used in the steady-state Navier-Stokes problem(cf.[6]).展开更多
This paper discusses the stability of theoretical solutions for nonlinear multi-variable delay perturbation problems (MVDPP) of the form x′(t)=f(x(t),x(t-τ 1(t)),...,x(t-τ m(t)),y(t),y(t-τ 1(t)),...,y(t-τ m(t...This paper discusses the stability of theoretical solutions for nonlinear multi-variable delay perturbation problems (MVDPP) of the form x′(t)=f(x(t),x(t-τ 1(t)),...,x(t-τ m(t)),y(t),y(t-τ 1(t)),...,y(t-τ m(t))), and εy′(t)=g(x(t),x(t-τ 1(t)),...,x(t-τ m(t)),y(t),y(t-τ 1(t)),...,y(t-τ m(t))), where 0<ε1. A sufficient condition of stability for the systems is obtained. Additionally we prove the numerical solutions of the implicit Euler method are stable under this condition.展开更多
Two kinds of wavelet-based elements have been constructed to analyze the stability of plates and shells and the static displacement of 3D elastic problems.The scaling functions of B-spline wavelet on the interval（BSW...Two kinds of wavelet-based elements have been constructed to analyze the stability of plates and shells and the static displacement of 3D elastic problems.The scaling functions of B-spline wavelet on the interval（BSWI） are employed as interpolating functions to construct plate and shell elements for stability analysis and 3D elastic elements for static mechanics analysis.The main advantages of BSWI scaling functions are the accuracy of B-spline functions approximation and various wavelet-based elements for structural analysis.The performances of the present elements are demonstrated by typical numerical examples.展开更多
Fourth order differential equations are considered to develop the class of methods for the numerical solution of boundary value problems. In this paper, we have discussed the regions of absolute stability of fourth or...Fourth order differential equations are considered to develop the class of methods for the numerical solution of boundary value problems. In this paper, we have discussed the regions of absolute stability of fourth order boundary value problems. Methods proposed and derived in this paper are applied to solve a fourth-order boundary value problem. Numerical results are given to illustrate the efficiency of our methods and compared with exact solution.展开更多
A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered. This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh si...A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered. This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h -- O（H2）, which can still maintain the asymptotically optimal accuracy. It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution, which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h. Hence, the two-level stabilized finite element method can save a large amount of computational time. Moreover, numerical tests confirm the theoretical results of the present method.展开更多
The paper gives two examples of larger construction projects with typical stability problems. The first example is from Sakhalin Island in the Russian Far East. It is connected with a construction of oil and gas pipel...The paper gives two examples of larger construction projects with typical stability problems. The first example is from Sakhalin Island in the Russian Far East. It is connected with a construction of oil and gas pipelines through the mountainous terrain in Makarov region. The region has an active geotectonic history and is highly affected by uncontrolled erosion and extensive landslips. Basic principles of landslide hazard mitigation are presented. The second example is from a motorway construction in Azerbaijan. This motorway leads from Baku to Russia through a seismo-tectonically active area at the toe of Caucasian mountains and in some places is situated in deep cuts at the toe of high slopes. This unsuitable routing, together with seismic activity, led to a slope stability failure of a slope affected by recent tectonic movements near the village of Devechi. Stability conditions and designed remedy measures are presented.展开更多
A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoot...A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoothing technologies.Some important properties of the proposed novel equivalent model are summarized.In addition,the stability properties of the proposed steepest descent-based neural networks for LCP are analyzed.In order to illustrate the theoretical results,we provide some numerical simulations and compare the proposed neural networks with existing neural networks based on the NCP-functions.Numerical results indicate that the performance of the proposed neural networks is effective and robust.展开更多
Two simplifled and stabilized mixed element formats for the Stokes problem are derived by bubble function, and their convergence, i.e., error analysis, are proved. These formats can save more freedom degrees than othe...Two simplifled and stabilized mixed element formats for the Stokes problem are derived by bubble function, and their convergence, i.e., error analysis, are proved. These formats can save more freedom degrees than other usual formats.展开更多
It is weN-known that the standard Galerkin is not ideally suited to deal with the spatial discretization of convection-dominated problems. In this paper, several techniques are proposed to overcome the instabilitY iss...It is weN-known that the standard Galerkin is not ideally suited to deal with the spatial discretization of convection-dominated problems. In this paper, several techniques are proposed to overcome the instabilitY issues in convection-dominated problems in the simulation with a meshless method. These stable techniques included nodal refinement, enlargement of the nodal influence domain, full upwind meshless technique and adaptive upwind meshless technique. Numerical results for sample problems show that these techniques are effective in solving convection-dominated problems, and the adaptive upwind meshless technique is the most effective method of all.展开更多
In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties...In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties. In this paper, we extend these results to general linear methods and to more generalproblem class Kστ. The concepts of (k, p, q)-secondary stability and (k, p. q)-secondary stability are introduced, and the criteria of secondary algebraic stability are also established. The criteria relax algebraicstability conditions while retaining the virtues of a nonlinear test problem.展开更多
Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they a...Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.展开更多
This paper extends the quantitative stability results to a more general class of two-stage stochastic variational inequality problems(TSVIP).The existence of solutions to the TSVIP is discussed,and the quantitative re...This paper extends the quantitative stability results to a more general class of two-stage stochastic variational inequality problems(TSVIP).The existence of solutions to the TSVIP is discussed,and the quantitative relationship between the TSVIP and its distribution perturbed problem is derived.展开更多
Based on the contact equivalent relation of smooth map-germs in singularity theory, the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameter...Based on the contact equivalent relation of smooth map-germs in singularity theory, the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameter symmetry is discussed. Some basic results are obtained. Transversality condition is used to characterize the stability of equavariant bifurcation problems.展开更多
This article is contributed to the Cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positiv...This article is contributed to the Cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positive solutions of △u ＋ K（｜x｜）u^p =0, is obtained when p is critical for general K（｜x｜）.展开更多
Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical p...Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.展开更多
This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditiona...This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai＇s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.展开更多
This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incr...This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incremental unknowns methods. Through the stability analyzing for the scheme, it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.展开更多
The symplectic approach proposed and developed by Zhong et al. in 1990s for elasticity problems is a rational analytical method, in which ample experience is not needed as in the conventional semi-inverse method. In t...The symplectic approach proposed and developed by Zhong et al. in 1990s for elasticity problems is a rational analytical method, in which ample experience is not needed as in the conventional semi-inverse method. In the symplectic space, elasticity problems can be solved using the method of separation of variables along with the eigenfunction expansion technique, as in traditional Fourier analysis. The eigensolutions include those corresponding to zero and nonzero eigenvalues. The latter group can be further divided into α-and β-sets. This paper reformulates the form of β-set eigensolutions to achieve the stability of numerical calculation, which is very important to obtain accurate results within the symplectic frame. An example is finally given and numerical results are compared and discussed.展开更多
In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). I...In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). Included are remarks on multiple solutions, multi-step methods, effect of initial value perturbations, as well as slowing and advancing the computed motion in second order problems.展开更多
文摘For stabilized saddle-point problems, we apply the two iteration parameters idea for regularized Hermitian and skew-Hermitian splitting (RHSS) method and establish accelerated RHSS (ARHSS) iteration method. Theoretical analysis shows that the ARHSS method converges unconditionally to the unique solution of the saddle point problem. Finally, we use a numerical example to confirm the effectiveness of the method.
文摘In this paper,we consider the so-called "inexact Uzawa" algorithm applied to the unstable Navier-Stokes problem.We use stabilization matrix to stabilize the unstable system and proved theoretically that under given proper preconditioners,Uzawa algorithm is convergent for the stablization system.Bounds for the iteration error are provided.We show numerically that Uzawa algorithm is convergent as well for the stabilization systems when it is used in the steady-state Navier-Stokes problem(cf.[6]).
文摘This paper discusses the stability of theoretical solutions for nonlinear multi-variable delay perturbation problems (MVDPP) of the form x′(t)=f(x(t),x(t-τ 1(t)),...,x(t-τ m(t)),y(t),y(t-τ 1(t)),...,y(t-τ m(t))), and εy′(t)=g(x(t),x(t-τ 1(t)),...,x(t-τ m(t)),y(t),y(t-τ 1(t)),...,y(t-τ m(t))), where 0<ε1. A sufficient condition of stability for the systems is obtained. Additionally we prove the numerical solutions of the implicit Euler method are stable under this condition.
基金supported by the National Natural Science Foundation of China (No. 50805028)the Key Project of Chinese Ministry of Education (No. 210170)+1 种基金Guangxi key Technologies R & D Program of China (Nos. 1099022-1 and 0900705 003)supported in part by the Excellent Talents in Guangxi Higher Education Institutions of China
文摘Two kinds of wavelet-based elements have been constructed to analyze the stability of plates and shells and the static displacement of 3D elastic problems.The scaling functions of B-spline wavelet on the interval（BSWI） are employed as interpolating functions to construct plate and shell elements for stability analysis and 3D elastic elements for static mechanics analysis.The main advantages of BSWI scaling functions are the accuracy of B-spline functions approximation and various wavelet-based elements for structural analysis.The performances of the present elements are demonstrated by typical numerical examples.
文摘Fourth order differential equations are considered to develop the class of methods for the numerical solution of boundary value problems. In this paper, we have discussed the regions of absolute stability of fourth order boundary value problems. Methods proposed and derived in this paper are applied to solve a fourth-order boundary value problem. Numerical results are given to illustrate the efficiency of our methods and compared with exact solution.
基金Project supported by the National Natural Science Foundation of China(Nos.10901131,10971166, and 10961024)the National High Technology Research and Development Program of China (No.2009AA01A135)the Natural Science Foundation of Xinjiang Uygur Autonomous Region (No.2010211B04)
文摘A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered. This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h -- O（H2）, which can still maintain the asymptotically optimal accuracy. It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution, which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h. Hence, the two-level stabilized finite element method can save a large amount of computational time. Moreover, numerical tests confirm the theoretical results of the present method.
文摘The paper gives two examples of larger construction projects with typical stability problems. The first example is from Sakhalin Island in the Russian Far East. It is connected with a construction of oil and gas pipelines through the mountainous terrain in Makarov region. The region has an active geotectonic history and is highly affected by uncontrolled erosion and extensive landslips. Basic principles of landslide hazard mitigation are presented. The second example is from a motorway construction in Azerbaijan. This motorway leads from Baku to Russia through a seismo-tectonically active area at the toe of Caucasian mountains and in some places is situated in deep cuts at the toe of high slopes. This unsuitable routing, together with seismic activity, led to a slope stability failure of a slope affected by recent tectonic movements near the village of Devechi. Stability conditions and designed remedy measures are presented.
基金Supported by the National Natural Science Foundation of China(12371378,41725017,11901098).
文摘A family of neural networks is proposed to solve linear complementarity problems(LCP).The neural networks are constructed from the novel equivalent model of LCP,which is reformulated by utilizing the modulus and smoothing technologies.Some important properties of the proposed novel equivalent model are summarized.In addition,the stability properties of the proposed steepest descent-based neural networks for LCP are analyzed.In order to illustrate the theoretical results,we provide some numerical simulations and compare the proposed neural networks with existing neural networks based on the NCP-functions.Numerical results indicate that the performance of the proposed neural networks is effective and robust.
文摘Two simplifled and stabilized mixed element formats for the Stokes problem are derived by bubble function, and their convergence, i.e., error analysis, are proved. These formats can save more freedom degrees than other usual formats.
基金the National Natural Science Foundation of China(No.10590353)theNatural Science Foundation of Shaanxi Province of China(No.2005A16)
文摘It is weN-known that the standard Galerkin is not ideally suited to deal with the spatial discretization of convection-dominated problems. In this paper, several techniques are proposed to overcome the instabilitY issues in convection-dominated problems in the simulation with a meshless method. These stable techniques included nodal refinement, enlargement of the nodal influence domain, full upwind meshless technique and adaptive upwind meshless technique. Numerical results for sample problems show that these techniques are effective in solving convection-dominated problems, and the adaptive upwind meshless technique is the most effective method of all.
文摘In 1992, Cooper [2] has presented some new stability concepts for Runge-Kutta methods whichis based on two slightly different test problems, and obtained the algebraic conditions that guarantee newstability properties. In this paper, we extend these results to general linear methods and to more generalproblem class Kστ. The concepts of (k, p, q)-secondary stability and (k, p. q)-secondary stability are introduced, and the criteria of secondary algebraic stability are also established. The criteria relax algebraicstability conditions while retaining the virtues of a nonlinear test problem.
文摘Two methods of stability analysis of systems described by dynamical equations are being considered. They are based on an analysis of eigenvalues spectrum for the evolutionary matrix or the spectral equation and they allow determining the conditions of stability and instability, as well as the possibility of chaotic behavior of systems in case of a stability loss. The methods are illustrated for nonlinear Lorenz and Rossler model problems.
基金Supported by the Guangxi Natural Science Foundation (2024GXNSFBA010345)the Innovation and Entrepreneurship Training Program of Guangxi Minzu University (S202310608001)。
文摘This paper extends the quantitative stability results to a more general class of two-stage stochastic variational inequality problems(TSVIP).The existence of solutions to the TSVIP is discussed,and the quantitative relationship between the TSVIP and its distribution perturbed problem is derived.
基金Project supported by the National Natural Science Foundation of China(No.10671002)the Natural Science Foundation of Hunan Province of China(No.04JJ3072)the Science Foundation of the Education Department of Hunan Province of China(No.04C383)
文摘Based on the contact equivalent relation of smooth map-germs in singularity theory, the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameter symmetry is discussed. Some basic results are obtained. Transversality condition is used to characterize the stability of equavariant bifurcation problems.
基金the National Natural Science Foundation of China(10471052,10631030)the PHD specialized grant of Ministry of Education of China(20060511001)
文摘This article is contributed to the Cauchy problem {δu/δt=△u＋K（｜x｜）u^p in R^n×（0,T）, u（x,0）=φ（x） in R^n;with initial function φ≡/0. The stability of positive radial steady state, which are positive solutions of △u ＋ K（｜x｜）u^p =0, is obtained when p is critical for general K（｜x｜）.
基金supported by the Key Disciplines of Shanghai Municipality (Operations Research & Cybernetics, No. S30104)Shanghai Leading Academic Discipline Project (No. J50101)
文摘Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.
基金The project supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China （G2000048702, 2003CB716707）the National Science Fund for Distinguished Young Scholars （10025208）+1 种基金 the National Natural Science Foundation of China （Key Program） （10532040） the Research Fund for 0versea Chinese （10228028）.
文摘This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai＇s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.
文摘This paper deals with the special nonlinear reaction-diffusion equation. The finite difference scheme with incremental unknowns approximating to the differential equation （2.1） is set up by means of introducing incremental unknowns methods. Through the stability analyzing for the scheme, it was shown that the stability conditions of the finite difference schemes with the incremental unknowns are greatly improved when compared with the stability conditions of the corresponding classic difference scheme.
基金the National Natural Science Foundation of China (Nos. 10725210 and 10432030) the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060335107)the Program for New Century Excellent Talents in University, MOE, China (No. NCET-05-05010)
文摘The symplectic approach proposed and developed by Zhong et al. in 1990s for elasticity problems is a rational analytical method, in which ample experience is not needed as in the conventional semi-inverse method. In the symplectic space, elasticity problems can be solved using the method of separation of variables along with the eigenfunction expansion technique, as in traditional Fourier analysis. The eigensolutions include those corresponding to zero and nonzero eigenvalues. The latter group can be further divided into α-and β-sets. This paper reformulates the form of β-set eigensolutions to achieve the stability of numerical calculation, which is very important to obtain accurate results within the symplectic frame. An example is finally given and numerical results are compared and discussed.
文摘In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). Included are remarks on multiple solutions, multi-step methods, effect of initial value perturbations, as well as slowing and advancing the computed motion in second order problems.
