The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent ...The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.展开更多
A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is ...A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.展开更多
A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector meth...A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.展开更多
The Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [ 1 ], and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2]. The methods have th...The Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [ 1 ], and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2]. The methods have the advantages of parallel computing, stability and good accuracy. Tn this paper for the two-dimensional diffusion equation, the net region is divided into bands, a special kind of block. This method is called the alternating Band Crank-Nicolson method.展开更多
In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods ...In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.展开更多
In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolso...In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.展开更多
A parallel diagonally iterated Runge Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the ...A parallel diagonally iterated Runge Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the iteration parameters of the method are tuned in such a way that fast convergence to the value of corrector is achieved.展开更多
Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. He...Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are rearranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.展开更多
In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the...In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nieolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.展开更多
In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme...In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numerical solution of initial and boundary value prob-lems for the one-dimension homogeneous wave equation. The initial deriva-tive condition is approximated by different second order difference quotients in order to examine which gives more accurate numerical results. The local truncation error, consistency and stability of the difference schemes CTCS, Crank-Nicolson and ω are also considered.展开更多
This paper proposes a continuous block method for the solution of second order ordinary differential equation. Collocation and interpolation of the power series approximate solution are adopted to derive a continuous ...This paper proposes a continuous block method for the solution of second order ordinary differential equation. Collocation and interpolation of the power series approximate solution are adopted to derive a continuous implicit linear multistep method. Continuous block method is used to derive the independent solution which is evaluated at selected grid points to generate the discrete block method. The order, consistency, zero stability and stability region are investigated. The new method was found to compare favourably with the existing methods in term of accuracy.展开更多
We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on...We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on nonlinear equations. We focus on to describe the analytic solution in the special pattern of travelling wave solutions using tan-cot function method. We discuss about inviscid and viscous version of Burger’s equation for fluid flow and investigate the effects of internal friction of a fluid via Reynolds number. By changing the velocity amplitude, the nature of flows with shock wave and disturbance are observed. For numerical solutions, the Crank-Nicolson scheme is introduced to establish the wave solutions.展开更多
The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off betwee...The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.展开更多
The main aim of this paper is to present and emphasize the contribution of stochastic numerical methods as must tools for the modern econometric modelisation. Indeed, the stochastic numerical methods play an important...The main aim of this paper is to present and emphasize the contribution of stochastic numerical methods as must tools for the modern econometric modelisation. Indeed, the stochastic numerical methods play an important role in mathematical modelling and the econometric analysis because they model uncertainties that govern the real-world data. However these powerful tools are not well-known and understood by many economists and financial econometricians.展开更多
文摘The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.
基金supported by the Yunnan Provincial Applied Basic Research Program of China(No. KKSY201207019)
文摘A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.
基金The Major State Research Program (G1999030803) of China and the NNSF (G10271066, 19972023) of China.
文摘A finite volume element predictor-corrector method for a class of nonlinear parabolic system of equations is presented and analyzed. Suboptimal L^2 error estimate for the finite volume element predictor-corrector method is derived. A numerical experiment shows that the numerical results are consistent with theoretical analysis.
基金The work presented in this paper was supported by the National Science Foundation of China
文摘The Alternating Segment Crank-Nicolson scheme for one-dimensional diffusion equation has been developed in [ 1 ], and the Alternating Block Crank-Nicolson method for two-dimensional problem in [2]. The methods have the advantages of parallel computing, stability and good accuracy. Tn this paper for the two-dimensional diffusion equation, the net region is divided into bands, a special kind of block. This method is called the alternating Band Crank-Nicolson method.
文摘In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.
基金supported by the Key Laboratory of Road Construction Technology and Equipment(Chang’an University,No.300102253502)the Natural Science Foundation of Shandong Province of China(GrantNo.ZR2022YQ06)the Development Plan of Youth Innovation Team in Colleges and Universities of Shandong Province(Grant No.2022KJ140).
文摘In this study,we propose an efficient numerical framework to attain the solution of the extended Fisher-Kolmogorov(EFK)problem.The temporal derivative in the EFK equation is approximated by utilizing the Crank-Nicolson scheme.Following temporal discretization,the generalized finite difference method(GFDM)with supplementary nodes is utilized to address the nonlinear boundary value problems at each time node.These supplementary nodes are distributed along the boundary to match the number of boundary nodes.By incorporating supplementary nodes,the resulting nonlinear algebraic equations can effectively satisfy the governing equation and boundary conditions of the EFK equation.To demonstrate the efficacy of our approach,we present three numerical examples showcasing its performance in solving this nonlinear problem.
文摘A parallel diagonally iterated Runge Kutta (PDIRK) method is constructed to solve stiff initial value problems for delay differential equations. The order and stability of this PDIRK method has been analyzed, and the iteration parameters of the method are tuned in such a way that fast convergence to the value of corrector is achieved.
文摘Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are rearranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.
基金in part supported by the Distinguished Young Scholars Fund of Xinjiang Province(2013711010)NCET-13-0988the NSF of China(11271313,11271298,61163027,and 11362021)
文摘In this paper, the Crank-Nicolson/Newton scheme for solving numerically second- order nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nieolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete Crank- Nicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.
文摘In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numerical solution of initial and boundary value prob-lems for the one-dimension homogeneous wave equation. The initial deriva-tive condition is approximated by different second order difference quotients in order to examine which gives more accurate numerical results. The local truncation error, consistency and stability of the difference schemes CTCS, Crank-Nicolson and ω are also considered.
文摘This paper proposes a continuous block method for the solution of second order ordinary differential equation. Collocation and interpolation of the power series approximate solution are adopted to derive a continuous implicit linear multistep method. Continuous block method is used to derive the independent solution which is evaluated at selected grid points to generate the discrete block method. The order, consistency, zero stability and stability region are investigated. The new method was found to compare favourably with the existing methods in term of accuracy.
文摘We study the nonlinear parabolic equations for travelling wave solutions of Burger’s equations. The purpose of the present work is to study various types of Burger’s equations describing waves and those are based on nonlinear equations. We focus on to describe the analytic solution in the special pattern of travelling wave solutions using tan-cot function method. We discuss about inviscid and viscous version of Burger’s equation for fluid flow and investigate the effects of internal friction of a fluid via Reynolds number. By changing the velocity amplitude, the nature of flows with shock wave and disturbance are observed. For numerical solutions, the Crank-Nicolson scheme is introduced to establish the wave solutions.
文摘The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.
文摘The main aim of this paper is to present and emphasize the contribution of stochastic numerical methods as must tools for the modern econometric modelisation. Indeed, the stochastic numerical methods play an important role in mathematical modelling and the econometric analysis because they model uncertainties that govern the real-world data. However these powerful tools are not well-known and understood by many economists and financial econometricians.
