In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG s...In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.展开更多
In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error esti...In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.展开更多
By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is propose...By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.展开更多
In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be r...In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.展开更多
By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differenti...By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].展开更多
In this paper we study the singularly penurbed boundary value problem: where e is a positive small parameter In the conditions: we prove the existences, and uniformly valid asymptotic expansions of solutions for the g...In this paper we study the singularly penurbed boundary value problem: where e is a positive small parameter In the conditions: we prove the existences, and uniformly valid asymptotic expansions of solutions for the given boundary value problems, and hence we improve the existing results.展开更多
A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary la...A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary layer, and he nce the study of asymptotic behavior of their solutions seems more diffcult. The uniformly valid asymptotic expansions of solutions as well as their derivatives are given via the upper and lower solutions method, and those estimates seem qu ite accurate.展开更多
We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The ac...We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.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
Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind sch...Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.展开更多
This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matr...This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.展开更多
In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal...In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.展开更多
In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving...In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.展开更多
This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transfo...This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations.We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework.The convergence rate is valid independent of the small parameter.Furthermore,we establish a sharper L^(2)-error estimate if the true solution has a special regular component.Numerical experiments are also given.展开更多
We construct a positive type difference scheme for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ? , to the solution...We construct a positive type difference scheme for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ? , to the solution of the singularly perturbed B. V.P. Numerical examples are provided.展开更多
The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the sp...The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.展开更多
In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order converge...In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.展开更多
Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly pertur...Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.展开更多
This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singula...This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.展开更多
基金Supported by the National Natural Science Foundation of China (10571053,10871066)the Programme for New Century Excellent Talents in University (NCET-06-0712).
文摘In this paper, we consider the local discontinuous Galerkin method （LDG） for solving singularly perturbed convection-diffusion problems in one- and two-dimensional settings. The existence and uniqueness of the LDG solutions are verified. Numerical experiments demonstrate that it seems impossible to obtain uniform superconvergence for numerical fluxes under uniform meshes. Thanks to the implementation of two-type different anisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform 2p ＋ i-order superconvergence is observed numerically for both one-dimensional and twodimensional cases.
文摘In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.
基金supported by the Natural Science Foundation of Zhejiang Province,China(Grant Nos.LY20A010021,LY19A010002,LY20G030025)the Natural Science Founda-tion of Ningbo City,China(Grant Nos.2021J147,2021J235).
文摘By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.
基金supported by the National Natural Science Foundation of China (No.12172154)the 111 Project (No.B14044)+1 种基金the Natural Science Foundation of Gansu Province (No.23JRRA1035)the Natural Science Foundation of Anhui University of Finance and Economics (No.ACKYC20043).
文摘In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.
基金Project supported by the National Natural Science Foundation of China.
文摘By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].
文摘In this paper we study the singularly penurbed boundary value problem: where e is a positive small parameter In the conditions: we prove the existences, and uniformly valid asymptotic expansions of solutions for the given boundary value problems, and hence we improve the existing results.
文摘A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary layer, and he nce the study of asymptotic behavior of their solutions seems more diffcult. The uniformly valid asymptotic expansions of solutions as well as their derivatives are given via the upper and lower solutions method, and those estimates seem qu ite accurate.
文摘We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.
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 Department of Science & Technology, Government of India under research grant SR/S4/MS:318/06.
文摘Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.
基金Project supported by the National Natural Science Foundation of China(No.10672194)the China-Russia Cooperative Project(the National Natural Science Foundation of China and the Russian Foundation for Basic Research)(No.10811120012)
文摘This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.
文摘In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.
基金the Council of Scientific and Industrial Research,New Delhi,India for its financial support.
文摘In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.
基金Supported by the National Natural Science Foundation of China(11801396)National College Students Innovation and Entrepreneurship Training Project(202210332019Z)。
文摘This paper concerns a discontinuous Galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations.We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework.The convergence rate is valid independent of the small parameter.Furthermore,we establish a sharper L^(2)-error estimate if the true solution has a special regular component.Numerical experiments are also given.
文摘We construct a positive type difference scheme for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ? , to the solution of the singularly perturbed B. V.P. Numerical examples are provided.
文摘The numerical solution of a singularly perturbed problem for the semilinear parabolic differential equation with parabolic boundary layers is discussed. A nonlinear two-level difference scheme is constructed on the special non-uniform grids. The uniform con vergence of this scheme is proved and some numerical examples are given.
文摘In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.
基金Project supported by the National Natural Science Foundation of China(Key Program)(Nos.11132004 and 51078145)
文摘Based on the precise integration method （PIM）, a coupling technique of the high order multiplication perturbation method （HOMPM） and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems （TPBVPs） with one boundary layer. First, the inhomogeneous ordinary differential equations （ODEs） are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.
基金supported by the National Natural Science Foundation of China(Key Program)(Nos.11132004 and 51078145)
文摘This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.
