Although predictor-corrector methods have been extensively applied,they might not meet the requirements of practical applications and engineering tasks,particularly when high accuracy and efficiency are necessary.A no...Although predictor-corrector methods have been extensively applied,they might not meet the requirements of practical applications and engineering tasks,particularly when high accuracy and efficiency are necessary.A novel class of correctors based on feedback-accelerated Picard iteration(FAPI)is proposed to further enhance computational performance.With optimal feedback terms that do not require inversion of matrices,significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts;however,the computational complexities are comparably low.These advantages enable nonlinear engineering problems to be solved quickly and accurately,even with rough initial guesses from elementary predictors.The proposed method offers flexibility,enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach.In our method,the functional formulas of FAPI are discretized into numerical forms using the collocation approach.These collocated iteration formulas can directly solve nonlinear problems,but they may require significant computational resources because of the manipulation of high-dimensionalmatrices.To address this,the collocated iteration formulas are further converted into finite difference forms,enabling the design of lightweight predictor-corrector algorithms for real-time computation.The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches:the modified Euler approach,the Adams-Bashforth-Moulton approach,and the implicit Runge-Kutta approach.Subsequently,the updated approaches are tested in solving strongly nonlinear problems,including the Matthieu equation,the Duffing equation,and the low-earth-orbit tracking problem.The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.展开更多
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
To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and invest...To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.展开更多
Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid...Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, V-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.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
We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponent...We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.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
In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain a...In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform.We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps.The variable V-cycle multigrid preconditioner are also available.展开更多
Panel methods for the calculation of wavemaking resistance result in a linear equation system for the unknown singularities.The coefficient matrix is full but not well conditioned.In this paper an incomplete LU decomp...Panel methods for the calculation of wavemaking resistance result in a linear equation system for the unknown singularities.The coefficient matrix is full but not well conditioned.In this paper an incomplete LU decomposition (ILU) method and a combined multigrid ILU method are used to solve the linear system.Systematic computations using the ILU method have shown that the CPU time can be reduced to 30% to 40% of that using an incomplete Gaussian elimination method. In the proposed multigrid ILU method an averaged restriction and a piecewise constant prolongation are used.The construction of the coefficient matrix at coarse levels is based on geometrical considerations.It turns out that the condition of the relative consistency is fulfilled.Comparison computations have shown that nearly the same results were obtained.However,due to additional CPU time needed for the execution of the matrix vector products in the restriction and the prolongation proceses of the multigrid method,a further reduction of the total CPU time could not be reailized.展开更多
In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.
A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-bal...A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.展开更多
In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and th...In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Usingerror norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.展开更多
This paper presents a numerical method for PDE-constrained optimization problems. These problems arise in many fields of science and engineering including those dealing with real applications. The physical problem is ...This paper presents a numerical method for PDE-constrained optimization problems. These problems arise in many fields of science and engineering including those dealing with real applications. The physical problem is modeled by partial differential equations (PDEs) and involve optimization of some quantity. The PDEs are in most cases nonlinear and solved using numerical methods. Since such numerical solutions are being used routinely, the recent trend has been to develop numerical methods and algorithms so that the optimization problems can be solved numerically as well using the same PDE-solver. We present here one such numerical method which is based on simultaneous pseudo-time stepping. The efficiency of the method is increased with the help of a multigrid strategy. Application example is included for an aerodynamic shape optimization problem.展开更多
The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consis...The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.展开更多
In this paper, an efficient multigrid fictitious boundary method （MFBM） coupled with the FEM solver package FEATFLOW was used for the detailed simulation of incompressible viscous flows around one or more moving NAC...In this paper, an efficient multigrid fictitious boundary method （MFBM） coupled with the FEM solver package FEATFLOW was used for the detailed simulation of incompressible viscous flows around one or more moving NACA0012 airfoils. The calculations were carded on a fixed multigrid finite element mesh on which fluid equations were satisfied everywhere, and the airfoils were allowed to move freely through the mesh. The MFBM was employed to treat interactions between the fluid and the airfoils The motion of the airfoils was modeled by Newton-Euler equations. Numerical results of experiments verify that this method provides an efficient way to simulate incompressible viscous flows around moving airfoils.展开更多
In this paper, a V-cycle multigrid method is presented for a Hermite rectangular element. By defining proper mesh-dependent inner product and transfer operator, we obtain its convergence property and the uniform conve...In this paper, a V-cycle multigrid method is presented for a Hermite rectangular element. By defining proper mesh-dependent inner product and transfer operator, we obtain its convergence property and the uniform convergence rate independent of mesh size and level are established.展开更多
This paper describes a way of solving the reservoir simulation pressure equation using mulligrid technique. The subroutine MG of four-grid method is presented. The result for 2-D two-phase problem is exactly the same ...This paper describes a way of solving the reservoir simulation pressure equation using mulligrid technique. The subroutine MG of four-grid method is presented. The result for 2-D two-phase problem is exactly the same as that of the SOR method and the CPU time is much less than that of the latter one.展开更多
In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids....In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.展开更多
Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and develo...Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.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
基金work is supported by the Fundamental Research Funds for the Central Universities(No.3102019HTQD014)of Northwestern Polytechnical UniversityFunding of National Key Laboratory of Astronautical Flight DynamicsYoung Talent Support Project of Shaanxi State.
文摘Although predictor-corrector methods have been extensively applied,they might not meet the requirements of practical applications and engineering tasks,particularly when high accuracy and efficiency are necessary.A novel class of correctors based on feedback-accelerated Picard iteration(FAPI)is proposed to further enhance computational performance.With optimal feedback terms that do not require inversion of matrices,significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts;however,the computational complexities are comparably low.These advantages enable nonlinear engineering problems to be solved quickly and accurately,even with rough initial guesses from elementary predictors.The proposed method offers flexibility,enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach.In our method,the functional formulas of FAPI are discretized into numerical forms using the collocation approach.These collocated iteration formulas can directly solve nonlinear problems,but they may require significant computational resources because of the manipulation of high-dimensionalmatrices.To address this,the collocated iteration formulas are further converted into finite difference forms,enabling the design of lightweight predictor-corrector algorithms for real-time computation.The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches:the modified Euler approach,the Adams-Bashforth-Moulton approach,and the implicit Runge-Kutta approach.Subsequently,the updated approaches are tested in solving strongly nonlinear problems,including the Matthieu equation,the Duffing equation,and the low-earth-orbit tracking problem.The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.
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.
文摘To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.
基金Projects(2006AA06Z105, 2007AA06Z134) supported by the National High-Tech Research and Development Program of ChinaProjects(2007, 2008) supported by China Scholarship Council (CSC)
文摘Based on the fact that 3-D model discretization by artificial could not always be successfully implemented especially for large-scaled problems when high accuracy and efficiency were required, a new adaptive multigrid finite element method was proposed. In this algorithm, a-posteriori error estimator was employed to generate adaptively refined mesh on a given initial mesh. On these iterative meshes, V-cycle based multigrid method was adopted to fast solve each linear equation with each initial iterative term interpolated from last mesh. With this error estimator, the unknowns were nearly optimally distributed on the final mesh which guaranteed the accuracy. The numerical results show that the multigrid solver is faster and more stable compared with ICCG solver. Meanwhile, the numerical results obtained from the final model discretization approximate the analytical solutions with maximal relative errors less than 1%, which remarkably validates this algorithm.
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 project supported by National Natural Science Foundation of China under Grant No.19902002
文摘We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams–Bashforth–Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.
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This research was supported by the National Natural Science Foundation of China under grant 10071015
文摘In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform.We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps.The variable V-cycle multigrid preconditioner are also available.
文摘Panel methods for the calculation of wavemaking resistance result in a linear equation system for the unknown singularities.The coefficient matrix is full but not well conditioned.In this paper an incomplete LU decomposition (ILU) method and a combined multigrid ILU method are used to solve the linear system.Systematic computations using the ILU method have shown that the CPU time can be reduced to 30% to 40% of that using an incomplete Gaussian elimination method. In the proposed multigrid ILU method an averaged restriction and a piecewise constant prolongation are used.The construction of the coefficient matrix at coarse levels is based on geometrical considerations.It turns out that the condition of the relative consistency is fulfilled.Comparison computations have shown that nearly the same results were obtained.However,due to additional CPU time needed for the execution of the matrix vector products in the restriction and the prolongation proceses of the multigrid method,a further reduction of the total CPU time could not be reailized.
基金The rescarch was supported by the Doctoral Point Foundation of chinese Universities and NSF
文摘In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.
基金Project supported by the National Natural Science Foundation of China(Nos.91330205and 11421101)the National Key Research and Development Program of China(No.2016YFB0200603)
文摘A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.
文摘In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Usingerror norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.
文摘This paper presents a numerical method for PDE-constrained optimization problems. These problems arise in many fields of science and engineering including those dealing with real applications. The physical problem is modeled by partial differential equations (PDEs) and involve optimization of some quantity. The PDEs are in most cases nonlinear and solved using numerical methods. Since such numerical solutions are being used routinely, the recent trend has been to develop numerical methods and algorithms so that the optimization problems can be solved numerically as well using the same PDE-solver. We present here one such numerical method which is based on simultaneous pseudo-time stepping. The efficiency of the method is increased with the help of a multigrid strategy. Application example is included for an aerodynamic shape optimization problem.
基金supported by the Natural Science Foundation of Hubei Province(CN)(Grant No.2019CFB693)the Research Foundation of the Education Department of Hubei Province(CN)(Grant No.B2019003)the open Foundation of the Key Laboratory of Metallurgical Equipment and Control of Education Ministry(CN)(Grant No.2015B14).
文摘The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.
基金Supported by National 863 Plan Project of Ministry of Science and Technology of China under Grant No. 2006AA09Z354National Natural Science Foundation of China under Grant No. 10672101.
文摘In this paper, an efficient multigrid fictitious boundary method （MFBM） coupled with the FEM solver package FEATFLOW was used for the detailed simulation of incompressible viscous flows around one or more moving NACA0012 airfoils. The calculations were carded on a fixed multigrid finite element mesh on which fluid equations were satisfied everywhere, and the airfoils were allowed to move freely through the mesh. The MFBM was employed to treat interactions between the fluid and the airfoils The motion of the airfoils was modeled by Newton-Euler equations. Numerical results of experiments verify that this method provides an efficient way to simulate incompressible viscous flows around moving airfoils.
基金Supported by NSF of China(10971203)Supported by the NSF of the education Department of Henan Province (2009A110017)
文摘In this paper, a V-cycle multigrid method is presented for a Hermite rectangular element. By defining proper mesh-dependent inner product and transfer operator, we obtain its convergence property and the uniform convergence rate independent of mesh size and level are established.
文摘This paper describes a way of solving the reservoir simulation pressure equation using mulligrid technique. The subroutine MG of four-grid method is presented. The result for 2-D two-phase problem is exactly the same as that of the SOR method and the CPU time is much less than that of the latter one.
文摘In this paper, a high order compact difference scheme and a multigrid method are proposed for solving two-dimensional (2D) elliptic problems with variable coefficients and interior/boundary layers on nonuniform grids. Firstly, the original equation is transformed from the physical domain (with a nonuniform mesh) to the computational domain (with a uniform mesh) by using a coordinate transformation. Then, a fourth order compact difference scheme is proposed to solve the transformed elliptic equation on uniform girds. After that, a multigrid method is employed to solve the linear algebraic system arising from the difference equation. At last, the numerical experiments on some elliptic problems with interior/boundary layers are conducted to show high accuracy and high efficiency of the present method.
文摘Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.
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.
