Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the conv...Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.展开更多
In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improv...In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.展开更多
For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods...For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .展开更多
In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed poin...In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.展开更多
The neutron shielding component of ITER （International Thermonuclear Experimental Reactor） vacuum vessel is a kind of structure resembling a wall in appearance. A FE （finite element） model is set up by using ANSYS...The neutron shielding component of ITER （International Thermonuclear Experimental Reactor） vacuum vessel is a kind of structure resembling a wall in appearance. A FE （finite element） model is set up by using ANSYS code in terms of its structural features. Static analysis, thermal expansion analysis and dynamic analysis are performed. The static results show that the stress and displacement distribution are allowable, but the high stress appears in the junction between the upper and lower parts. The modal analysis indicates that the biggest deformation exists in the port area. Through modal superposition, the single-point response has been found with the lower rank frequency of the acceleration seismic response spectrum. But the deformation and the stress values are within the permissible limit. The analysis results would benefit the work in the next step and provide some reference for the implementation of the engineering plan in the future.展开更多
The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and emp...The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.展开更多
For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a sm...For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.展开更多
A new iterating method based on homotopy function is developed in this paper. All solutions can be found easily without the need of choosing proper initial values. Compared to the homotopy continuation method, the sol...A new iterating method based on homotopy function is developed in this paper. All solutions can be found easily without the need of choosing proper initial values. Compared to the homotopy continuation method, the solution process of the present method is simplified, and the computation efficiency as well as the reliability for obtaining all solutions is also improved. By application of the method to the mechanisms problems, the results are satisfactory.展开更多
Iteration methods and their convergences of the maximum likelihoodestimator are discussed in this paper.We study Gauss-Newton method and give a set ofsufficient conditions for the convergence of asymptotic numerical s...Iteration methods and their convergences of the maximum likelihoodestimator are discussed in this paper.We study Gauss-Newton method and give a set ofsufficient conditions for the convergence of asymptotic numerical stability.The modifiedGauss-Newton method is also studied and the sufficient conditions of the convergence arepresented.Two numerical examples are given to illustrate our results.展开更多
The parallel algorithms of iterated defect correction methods (PIDeCM’s) are constructed, which are of efficiency and high order B-convergence for general nonlinear stiff systems in ODE’S. As the basis of constructi...The parallel algorithms of iterated defect correction methods (PIDeCM’s) are constructed, which are of efficiency and high order B-convergence for general nonlinear stiff systems in ODE’S. As the basis of constructing and discussing PIDeCM’s. a class of parallel one-leg methods is also investigated, which are of particular efficiency for linear systems.展开更多
By using the asymptotic iteration method, we have calculated numerically the eigenvalues En of the hyperbolic single wave potential which is introduced by H. Bahlouli, and A. D. Alhaidari. They found a new approach (t...By using the asymptotic iteration method, we have calculated numerically the eigenvalues En of the hyperbolic single wave potential which is introduced by H. Bahlouli, and A. D. Alhaidari. They found a new approach (the “potential parameter” approach) which has been adopted for this eigenvalues problem. For a fixed energy, the problem is solvable for a set of values of the potential parameters (the “parameter spectrum”). This paper will introduce a related work to complete the goal of finding the eigenvalues, the Schr?dinger equation with hyperbolic single wave potential is solved by using asymptotic iteration method. It is found that asymptotically this method gives accurate results for arbitrary parameters, V0, γ, and λ.展开更多
The finite element method is a key player in computational electromag-netics for designing RF(Radio Frequency)components such as waveguides.The frequency-domain analysis is fundamental to identify the characteristics ...The finite element method is a key player in computational electromag-netics for designing RF(Radio Frequency)components such as waveguides.The frequency-domain analysis is fundamental to identify the characteristics of the components.For the conventional frequency-domain electromagnetic analysis using FEM(Finite Element Method),the system matrix is complex-numbered as well as indefinite.The iterative solvers can be faster than the direct solver when the solver convergence is guaranteed and done in a few steps.However,such complex-numbered and indefinite systems are hard to exploit the merit of the iterative solver.It is also hard to benefit from matrix factorization techniques due to varying system matrix parts according to frequency.Overall,it is hard to adopt conventional iterative solvers even though the system matrix is sparse.A new parallel iterative FEM solver for frequency domain analysis is implemented for inhomogeneous waveguide structures in this paper.In this implementation,the previous solution of the iterative solver of Matlab(Matrix Laboratory)employ-ing the preconditioner is used for the initial guess for the next step’s solution process.The overlapped parallel stage using Matlab’s Parallel Computing Toolbox is also proposed to alleviate the cold starting,which ruins the convergence of early steps in each parallel stage.Numerical experiments based on waveguide structures have demonstrated the accuracy and efficiency of the proposed scheme.展开更多
For a specific combustion problem involving calculations of several species at the equilibrium state, it is simpler to write a general computer program and calculate the combustion concentration. Original work describ...For a specific combustion problem involving calculations of several species at the equilibrium state, it is simpler to write a general computer program and calculate the combustion concentration. Original work describes, an adaptation of Newton-Raphson method was used for solving the highly nonlinear system of equations describing the formation of equilibrium products in reacting of fuel-additive-air mixtures. This study also shows what possible of the results. In this paper, to be present the efficient numerical algorithms for. solving the combustion problem, to be used nonlinear equations based on the iteration method and high order of the Taylor series. The modified Adomian decomposition method was applied to construct the numerical algorithms. Some numerical illustrations are given to show the efficiency of algorithms. Comparisons of results by the new Matlab routines and previous routines, the result data indicate that the new Matlab routines are reliable, typical deviations from previous results are less than 0.05%.展开更多
In this paper, we aim to solve a two compartmental mathematical model of ordinary differential equations for cardiovascular-respiratory system using a new recent method: Perturbation Iteration method. The description ...In this paper, we aim to solve a two compartmental mathematical model of ordinary differential equations for cardiovascular-respiratory system using a new recent method: Perturbation Iteration method. The description of this method for different order derivatives in the Taylor Series expansion is discussed. This method provides the solution in the form of an infinite series for ordinary differential equation. The efficiency of the method used is investigated by a comparison of Euler method and Runge Kutta. Numerical simulations of all these three methods are implemented in Matlab. The validation has been carried out by taking the values of determinant parameters of cardiovascular-respiratory system for a 30 years old woman who is supposed to make practice of three regular physical activities: Walking, Jogging and Running fast. The results are in good agreement with experimental data.展开更多
Equivalent staggered-grid(ESG) as a new family of schemes has been utilized in seismic modeling,imaging,and inversion.Traditionally,the Taylor series expansion is often applied to calculate finite-difference(FD) coeff...Equivalent staggered-grid(ESG) as a new family of schemes has been utilized in seismic modeling,imaging,and inversion.Traditionally,the Taylor series expansion is often applied to calculate finite-difference(FD) coefficients on spatial derivatives,but the simulation results suffer serious numerical dispersion on a large frequency zone.We develop an optimized equivalent staggered-grid(OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3 D elastic wave equation.On the one hand,we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave,S-wave,and converted-wave(C-wave) terms.On the other hand,a novel plane wave solution for the 3 D elastic wave equation is derived from the matrix decomposition method to construct the time-space dispersion relations.FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method.Finally,we construct a new objective function to analyze P-wave,S-wave,and C-wave dispersion concerning frequencies.The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method.The synthetic examples demonstrate the effectiveness and superiority of the presented method.展开更多
A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots ...A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.展开更多
Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared ...Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared with the standard scattering problem for one obstacle, the difficulty of such a new problem is the interaction of scattered wave by different obstacles. A decomposition method for solving this multiple scattering problem is developed. Using the boundary integral equation method, we decompose the total scattered field into a sum of contributions by separated obstacles. Each contribution corresponds to scattering problem of single obstacle. However, all the single scattering problems are coupled via the boundary conditions, representing the physical interaction of scattered wave by different obstacles. We prove the feasibility of such a decomposition. To compute these contributions efficiently, an iteration algorithm of Jacobi type is proposed, decoupling the interaction of scattered wave from the numerical points of view. Under the well-separation assumptions on multiple obstacles, we prove the convergence of iteration sequence generated by the Jacobi algorithm, and give the error estimate between exact scattered wave and the iteration solution in terms of the obstacle size and the minimal distance of multiple obstacles. Such a quantitative description reveals the essences of wave scattering by multiple obstacles. Numerical examples showing the accuracy and convergence of our method are presented.展开更多
In this study, for the purpose of improving the efficiency and accuracy of numerical simulation of massive concrete, the symmetric successive over relaxation-preconditioned conjugate gradient method （SSOR-PCGM） with...In this study, for the purpose of improving the efficiency and accuracy of numerical simulation of massive concrete, the symmetric successive over relaxation-preconditioned conjugate gradient method （SSOR-PCGM） with an improved iteration format was derived and applied to solution of large sparse symmetric positive definite linear equations in the computational process of the finite element analysis. A three-dimensional simulation program for massive concrete was developed based on SSOR-PCGM with an improved iteration format. Then, the programs based on the direct method and SSOR-PCGM with an improved iteration format were used for computation of the Guandi roller compacted concrete （RCC） gravity dam and an elastic cube under free expansion. The comparison and analysis of the computational results show that SSOR-PCGM with the improved iteration format occupies much less physical memory and can solve larger-scale problems with much less computing time and flexible control of accuracy.展开更多
In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quan...In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence. We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this new method.展开更多
文摘Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.
文摘In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.
文摘For solving nonlinear and transcendental equation f(x)=0 , a singnificant improvement on Newton's method is proposed in this paper. New “Newton Like” methods are founded on the basis of Liapunov's methods of dynamic system. These new methods preserve quadratic convergence and computational efficiency of Newton's method, and remove the monotoneity condition imposed on f(x):f′(x)≠0 .
文摘In this paper we propose two original iterated maps to numerically approximate the nth root of a real number. Comparisons between the new maps and the famous Newton-Raphson method are carried out, including fixed point determination, stability analysis and measure of the mean convergence time, which is confirmed by our analytical convergence time model. Stability of solutions is confirmed by measuring the Lyapunov exponent over the parameter space of each map. A generalization of the second map is proposed, giving rise to a family of new maps to address the same problem. This work is developed within the language of discrete dynamical systems.
基金the National 973 program of China(No.2004CB720704)
文摘The neutron shielding component of ITER （International Thermonuclear Experimental Reactor） vacuum vessel is a kind of structure resembling a wall in appearance. A FE （finite element） model is set up by using ANSYS code in terms of its structural features. Static analysis, thermal expansion analysis and dynamic analysis are performed. The static results show that the stress and displacement distribution are allowable, but the high stress appears in the junction between the upper and lower parts. The modal analysis indicates that the biggest deformation exists in the port area. Through modal superposition, the single-point response has been found with the lower rank frequency of the acceleration seismic response spectrum. But the deformation and the stress values are within the permissible limit. The analysis results would benefit the work in the next step and provide some reference for the implementation of the engineering plan in the future.
文摘The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.
基金This study was co-supported by the National Key Research and Development Program of China(No.2021YFA0717100)the National Natural Science Foundation of China(Nos.12072270,U2013206).
文摘For over half a century,numerical integration methods based on finite difference,such as the Runge-Kutta method and the Euler method,have been popular and widely used for solving orbit dynamic problems.In general,a small integration step size is always required to suppress the increase of the accumulated computation error,which leads to a relatively slow computation speed.Recently,a collocation iteration method,approximating the solutions of orbit dynamic problems iteratively,has been developed.This method achieves high computation accuracy with extremely large step size.Although efficient,the collocation iteration method suffers from two limitations:(A)the computational error limit of the approximate solution is not clear;(B)extensive trials and errors are always required in tuning parameters.To overcome these problems,the influence mechanism of how the dynamic problems and parameters affect the error limit of the collocation iteration method is explored.On this basis,a parameter adjustment method known as the“polishing method”is proposed to improve the computation speed.The method proposed is demonstrated in three typical orbit dynamic problems in aerospace engineering:a low Earth orbit propagation problem,a Molniya orbit propagation problem,and a geostationary orbit propagation problem.Numerical simulations show that the proposed polishing method is faster and more accurate than the finite-difference-based method and the most advanced collocation iteration method.
文摘A new iterating method based on homotopy function is developed in this paper. All solutions can be found easily without the need of choosing proper initial values. Compared to the homotopy continuation method, the solution process of the present method is simplified, and the computation efficiency as well as the reliability for obtaining all solutions is also improved. By application of the method to the mechanisms problems, the results are satisfactory.
文摘Iteration methods and their convergences of the maximum likelihoodestimator are discussed in this paper.We study Gauss-Newton method and give a set ofsufficient conditions for the convergence of asymptotic numerical stability.The modifiedGauss-Newton method is also studied and the sufficient conditions of the convergence arepresented.Two numerical examples are given to illustrate our results.
文摘The parallel algorithms of iterated defect correction methods (PIDeCM’s) are constructed, which are of efficiency and high order B-convergence for general nonlinear stiff systems in ODE’S. As the basis of constructing and discussing PIDeCM’s. a class of parallel one-leg methods is also investigated, which are of particular efficiency for linear systems.
文摘By using the asymptotic iteration method, we have calculated numerically the eigenvalues En of the hyperbolic single wave potential which is introduced by H. Bahlouli, and A. D. Alhaidari. They found a new approach (the “potential parameter” approach) which has been adopted for this eigenvalues problem. For a fixed energy, the problem is solvable for a set of values of the potential parameters (the “parameter spectrum”). This paper will introduce a related work to complete the goal of finding the eigenvalues, the Schr?dinger equation with hyperbolic single wave potential is solved by using asymptotic iteration method. It is found that asymptotically this method gives accurate results for arbitrary parameters, V0, γ, and λ.
基金supported by Institute of Information&communications Technology Planning&Evaluation(ITP)grant funded by the Korea govermment(MSIT)(No.2019-0-00098,Advanced and Integrated Software Development for Electromagnetic Analysis)supported by Research Assistance Program(2021)in the Incheon National University.
文摘The finite element method is a key player in computational electromag-netics for designing RF(Radio Frequency)components such as waveguides.The frequency-domain analysis is fundamental to identify the characteristics of the components.For the conventional frequency-domain electromagnetic analysis using FEM(Finite Element Method),the system matrix is complex-numbered as well as indefinite.The iterative solvers can be faster than the direct solver when the solver convergence is guaranteed and done in a few steps.However,such complex-numbered and indefinite systems are hard to exploit the merit of the iterative solver.It is also hard to benefit from matrix factorization techniques due to varying system matrix parts according to frequency.Overall,it is hard to adopt conventional iterative solvers even though the system matrix is sparse.A new parallel iterative FEM solver for frequency domain analysis is implemented for inhomogeneous waveguide structures in this paper.In this implementation,the previous solution of the iterative solver of Matlab(Matrix Laboratory)employ-ing the preconditioner is used for the initial guess for the next step’s solution process.The overlapped parallel stage using Matlab’s Parallel Computing Toolbox is also proposed to alleviate the cold starting,which ruins the convergence of early steps in each parallel stage.Numerical experiments based on waveguide structures have demonstrated the accuracy and efficiency of the proposed scheme.
文摘For a specific combustion problem involving calculations of several species at the equilibrium state, it is simpler to write a general computer program and calculate the combustion concentration. Original work describes, an adaptation of Newton-Raphson method was used for solving the highly nonlinear system of equations describing the formation of equilibrium products in reacting of fuel-additive-air mixtures. This study also shows what possible of the results. In this paper, to be present the efficient numerical algorithms for. solving the combustion problem, to be used nonlinear equations based on the iteration method and high order of the Taylor series. The modified Adomian decomposition method was applied to construct the numerical algorithms. Some numerical illustrations are given to show the efficiency of algorithms. Comparisons of results by the new Matlab routines and previous routines, the result data indicate that the new Matlab routines are reliable, typical deviations from previous results are less than 0.05%.
文摘In this paper, we aim to solve a two compartmental mathematical model of ordinary differential equations for cardiovascular-respiratory system using a new recent method: Perturbation Iteration method. The description of this method for different order derivatives in the Taylor Series expansion is discussed. This method provides the solution in the form of an infinite series for ordinary differential equation. The efficiency of the method used is investigated by a comparison of Euler method and Runge Kutta. Numerical simulations of all these three methods are implemented in Matlab. The validation has been carried out by taking the values of determinant parameters of cardiovascular-respiratory system for a 30 years old woman who is supposed to make practice of three regular physical activities: Walking, Jogging and Running fast. The results are in good agreement with experimental data.
文摘Equivalent staggered-grid(ESG) as a new family of schemes has been utilized in seismic modeling,imaging,and inversion.Traditionally,the Taylor series expansion is often applied to calculate finite-difference(FD) coefficients on spatial derivatives,but the simulation results suffer serious numerical dispersion on a large frequency zone.We develop an optimized equivalent staggered-grid(OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3 D elastic wave equation.On the one hand,we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave,S-wave,and converted-wave(C-wave) terms.On the other hand,a novel plane wave solution for the 3 D elastic wave equation is derived from the matrix decomposition method to construct the time-space dispersion relations.FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method.Finally,we construct a new objective function to analyze P-wave,S-wave,and C-wave dispersion concerning frequencies.The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method.The synthetic examples demonstrate the effectiveness and superiority of the presented method.
基金Foundation item： Supported by the National Science Foundation of China（10701066） Supported by the National Foundation of the Education Department of Henan Province（2008A110022）
文摘A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.
基金supported by NSFC (11071039,11161130002)Natural Science Foundation of Jiangsu Province (BK2011584)
文摘Consider acoustic wave scattering by multiple obstacles with different sound properties on the boundary, which can be modeled by a mixed boundary value problem for the Helmholtz equation in frequency domain. Compared with the standard scattering problem for one obstacle, the difficulty of such a new problem is the interaction of scattered wave by different obstacles. A decomposition method for solving this multiple scattering problem is developed. Using the boundary integral equation method, we decompose the total scattered field into a sum of contributions by separated obstacles. Each contribution corresponds to scattering problem of single obstacle. However, all the single scattering problems are coupled via the boundary conditions, representing the physical interaction of scattered wave by different obstacles. We prove the feasibility of such a decomposition. To compute these contributions efficiently, an iteration algorithm of Jacobi type is proposed, decoupling the interaction of scattered wave from the numerical points of view. Under the well-separation assumptions on multiple obstacles, we prove the convergence of iteration sequence generated by the Jacobi algorithm, and give the error estimate between exact scattered wave and the iteration solution in terms of the obstacle size and the minimal distance of multiple obstacles. Such a quantitative description reveals the essences of wave scattering by multiple obstacles. Numerical examples showing the accuracy and convergence of our method are presented.
基金supported by the National Natural Science Foundation of China (Grant No.50808066)
文摘In this study, for the purpose of improving the efficiency and accuracy of numerical simulation of massive concrete, the symmetric successive over relaxation-preconditioned conjugate gradient method （SSOR-PCGM） with an improved iteration format was derived and applied to solution of large sparse symmetric positive definite linear equations in the computational process of the finite element analysis. A three-dimensional simulation program for massive concrete was developed based on SSOR-PCGM with an improved iteration format. Then, the programs based on the direct method and SSOR-PCGM with an improved iteration format were used for computation of the Guandi roller compacted concrete （RCC） gravity dam and an elastic cube under free expansion. The comparison and analysis of the computational results show that SSOR-PCGM with the improved iteration format occupies much less physical memory and can solve larger-scale problems with much less computing time and flexible control of accuracy.
文摘In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence. We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this new method.
