In this paper, we introduce a new extrapolation formula by combining Richardson extrapolation and Sloan iteration algorithms. Using this extrapolation formula, we obtain some asymptotic expansions of the Galerkin fini...In this paper, we introduce a new extrapolation formula by combining Richardson extrapolation and Sloan iteration algorithms. Using this extrapolation formula, we obtain some asymptotic expansions of the Galerkin finite element method for semi-simple eigenvalue problems of Fredholm integral equations of the second kind and improve the accuracy of the numerical approximations of the corresponding eigenvalues. Some numerical experiments ave carried out to demonstrate the effectiveness of our new method and to confirm our theoretical results.展开更多
With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.T...With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.Three extrapolation formulas are presented,whose temporal convergence orders in L_(∞)-norm are proved to be 2,3-α,and 4-2α,respectively,where 0<α<1.Similarly,by the method of order reduction,an extrapola-tion method is constructed for the fractional wave equation including two extrapolation formulas,which achieve temporal 4-γ and 6-2γ order in L_(∞)-norm,respectively,where1<γ<2.Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation,the fast extrapolation methods are obtained which reduce the computational complexity significantly while keep-ing the accuracy.Several numerical experiments confirm the theoretical results.展开更多
From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations （BIEs） with the logarithmic singularity. In this paper, mechanical quadrature methods （MQMs） are...From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations （BIEs） with the logarithmic singularity. In this paper, mechanical quadrature methods （MQMs） are presented to obtain the eigensolutions that are used to solve Laplace＇s equations. The MQMs possess high accuracy and low computation complexity. The convergence and the stability are proved based on Anselone＇s collective and asymptotical compact theory. An asymptotic expansion with odd powers of the errors is presented. By the h3-Richardson extrapolation algorithm （EA）, the accuracy order of the approximation can be greatly improved, and an a posteriori error estimate can be obtained as the self-adaptive algorithms. The efficiency of the algorithm is illustrated by examples.展开更多
An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in ...An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.展开更多
A new formula with derivatives for numerical integration was presented. Based on this formula and the Richardson extrapolafion process, a numerical integration method was established. It can converge faster than the R...A new formula with derivatives for numerical integration was presented. Based on this formula and the Richardson extrapolafion process, a numerical integration method was established. It can converge faster than the Romberg＇s. With the same accuracy, the computation of the new numerical integration with derivatives is only half of that of Romberg＇s numerical integration.展开更多
The steady, asymmetric and two-dimensional flow of viscous, incompressible and Newtonian fluid through a rectangular channel with splitter plate parallel to walls is investigated numerically. Earlier, the position of ...The steady, asymmetric and two-dimensional flow of viscous, incompressible and Newtonian fluid through a rectangular channel with splitter plate parallel to walls is investigated numerically. Earlier, the position of the splitter plate was taken as a centreline of channel but here it is considered its different positions which cause the asymmetric behaviour of the flow field. The geometric parameter that controls the position of splitter is defined as splitter position parameter a. The plane Poiseuille flow is considered far from upstream and downstream of the splitter. This flow-problem is solved numerically by a numerical scheme comprising a fourth order method, followed by a special finite-method. This numerical scheme transforms the governing equations to system of finite-difference equations, which are solved by point S.O.R. iterative method. In addition, the results obtained are further refined and upgraded by Richardson Extrapolation method. The calculations are carried out for the ranges -1 α R < 10<sup>5</sup>. The results are compared with existing literature regarding the symmetric case (when a = 0) for velocity, vorticity and skin friction distributions. The comparison is very favourable. Moreover, the notable thing is that the decay of vorticity to its downstream value takes place over an increasingly longer scale of x as R increases for symmetric case but it is not so for asymmetric one.展开更多
The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a uni...The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.展开更多
For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the...For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.展开更多
The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to ...The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to solve the timedependent problems parallel in time.This algorithm has received much interest from many researchers in the past years.We present in this paper a new variant of the parareal algorithm,which is derived by combining the original parareal algorithm and the Richardson extrapolation,for the numerical solution of the nonlinear ODEs and PDEs.Several nonlinear problems are tested to show the advantage of the new algorithm.The accuracy of the obtained numerical solution is compared with that of its original version(i.e.,the parareal algorithm based on the same numerical method).展开更多
Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to g...Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.展开更多
This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a ...This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method.Recursive application of this idea results in the EXCMG method proposed in this paper.Finally,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.展开更多
This paper deals with numerical computation and analysis for the initial boundary problems of two dimensional(2D)Sobolev equations with piecewise continuous argument.Firstly,a two-level high-order compact difference m...This paper deals with numerical computation and analysis for the initial boundary problems of two dimensional(2D)Sobolev equations with piecewise continuous argument.Firstly,a two-level high-order compact difference method(HOCDM)with computational accuracy O(τ^(2)+h_(x)^(4)+h_(y)^(4))is suggested,whereτ,h_(x),h_(y) denote the temporal and spatial stepsizes of the method,respectively.In order to improve the temporal computational accuracy of this method,the Richardson extrapolation technique is used and thus a new two-level HOCDMis derived,which is proved to be convergent of order four both in time and space.Although the new two-level HOCDM has the higher computational accuracy in time than the previous one,it will bring a larger computational cost.To overcome this deficiency,a three-level HOCDM with computational accuracy O(τ^(4)+h_(x)^(4)+h_(y)^(4))is constructed.Finally,with a series of numerical experiments,the theoretical accuracy and computational efficiency of the above methods are further verified.展开更多
This paper applies a difference scheme to a singularly perturbed problem. The authors provide two algorithms on moving mesh methods by using Richardson extrapolation which can improve the accuracy of numerical solutio...This paper applies a difference scheme to a singularly perturbed problem. The authors provide two algorithms on moving mesh methods by using Richardson extrapolation which can improve the accuracy of numerical solution. In traditional algorithms of moving meshes, the initial mesh is a uniform mesh. The authors change it to Bakhvalov-Shishkin mesh, and prove that it improves efficiency by numerical experiments. Finally, the results of the two algorithms are analyzed.展开更多
基金the Governor's Special Foundation of Guizhou Province for Outstanding Scientific Education Personnel (No.[2005]155),China
文摘In this paper, we introduce a new extrapolation formula by combining Richardson extrapolation and Sloan iteration algorithms. Using this extrapolation formula, we obtain some asymptotic expansions of the Galerkin finite element method for semi-simple eigenvalue problems of Fredholm integral equations of the second kind and improve the accuracy of the numerical approximations of the corresponding eigenvalues. Some numerical experiments ave carried out to demonstrate the effectiveness of our new method and to confirm our theoretical results.
基金supported by the National Natural Science Foundation of China(grant number 11671081).
文摘With the help of the asymptotic expansion for the classic Li formula and based on the L1-type compact difference scheme,we propose a temporal Richardson extrapolation method for the fractional sub-diffusion equation.Three extrapolation formulas are presented,whose temporal convergence orders in L_(∞)-norm are proved to be 2,3-α,and 4-2α,respectively,where 0<α<1.Similarly,by the method of order reduction,an extrapola-tion method is constructed for the fractional wave equation including two extrapolation formulas,which achieve temporal 4-γ and 6-2γ order in L_(∞)-norm,respectively,where1<γ<2.Combining the derived extrapolation methods with the fast algorithm for Caputo fractional derivative based on the sum-of-exponential approximation,the fast extrapolation methods are obtained which reduce the computational complexity significantly while keep-ing the accuracy.Several numerical experiments confirm the theoretical results.
基金Project supported by the National Natural Science Foundation of China (No. 10871034)
文摘From the potential theorem, the fundamental boundary eigenproblems can be converted into boundary integral equations （BIEs） with the logarithmic singularity. In this paper, mechanical quadrature methods （MQMs） are presented to obtain the eigensolutions that are used to solve Laplace＇s equations. The MQMs possess high accuracy and low computation complexity. The convergence and the stability are proved based on Anselone＇s collective and asymptotical compact theory. An asymptotic expansion with odd powers of the errors is presented. By the h3-Richardson extrapolation algorithm （EA）, the accuracy order of the approximation can be greatly improved, and an a posteriori error estimate can be obtained as the self-adaptive algorithms. The efficiency of the algorithm is illustrated by examples.
基金supported by the National Natural Science Foundation of China(No.51174236)the National Basic Research Program of China(973 Program)(No.2011CB606306)the Opening Project of State Key Laboratory of Porous Metal Materials(No.PMM-SKL-4-2012)
文摘An efficient high-order immersed interface method （IIM） is proposed to solve two-dimensional （2D） heat problems with fixed interfaces on Cartesian grids, which has the fourth-order accuracy in the maximum norm in both time and space directions. The space variable is discretized by a high-order compact （HOC） difference scheme with correction terms added at the irregular points. The time derivative is integrated by a Crank-Nicolson and alternative direction implicit （ADI） scheme. In this case, the time accuracy is just second-order. The Richardson extrapolation method is used to improve the time accuracy to fourth-order. The numerical results confirm the convergence order and the efficiency of the method.
文摘A new formula with derivatives for numerical integration was presented. Based on this formula and the Richardson extrapolafion process, a numerical integration method was established. It can converge faster than the Romberg＇s. With the same accuracy, the computation of the new numerical integration with derivatives is only half of that of Romberg＇s numerical integration.
文摘The steady, asymmetric and two-dimensional flow of viscous, incompressible and Newtonian fluid through a rectangular channel with splitter plate parallel to walls is investigated numerically. Earlier, the position of the splitter plate was taken as a centreline of channel but here it is considered its different positions which cause the asymmetric behaviour of the flow field. The geometric parameter that controls the position of splitter is defined as splitter position parameter a. The plane Poiseuille flow is considered far from upstream and downstream of the splitter. This flow-problem is solved numerically by a numerical scheme comprising a fourth order method, followed by a special finite-method. This numerical scheme transforms the governing equations to system of finite-difference equations, which are solved by point S.O.R. iterative method. In addition, the results obtained are further refined and upgraded by Richardson Extrapolation method. The calculations are carried out for the ranges -1 α R < 10<sup>5</sup>. The results are compared with existing literature regarding the symmetric case (when a = 0) for velocity, vorticity and skin friction distributions. The comparison is very favourable. Moreover, the notable thing is that the decay of vorticity to its downstream value takes place over an increasingly longer scale of x as R increases for symmetric case but it is not so for asymmetric one.
基金supported by the National Natural Science Foundation of China(Grant No.42274101)X.X.Wu was supported by the Fundamental Research Funds for the Central Universities of Central South University(Grant No.2020zzts354)+2 种基金H.L.Hu was supported by the National Natural Science Foundation of China(Grant No.12071128)by the Natural Science Foundation of Hunan Province(Grant No.2021JJ30434)Z.L.Li was supported by a Simons Grant No.633724.
文摘The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.
基金supported by National Natural Science Foundation of China(Grant Nos.11226332,41204082 and 11071067)the China Postdoctoral Science Foundation(Grant No.2011M501295)+1 种基金the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162120036)the Construct Program of the Key Discipline in Hunan Province
文摘For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.
基金This work was supported by NSF of China(Nos.10671078,60773195)and by Program for NCETthe State Education Ministry of China.
文摘The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to solve the timedependent problems parallel in time.This algorithm has received much interest from many researchers in the past years.We present in this paper a new variant of the parareal algorithm,which is derived by combining the original parareal algorithm and the Richardson extrapolation,for the numerical solution of the nonlinear ODEs and PDEs.Several nonlinear problems are tested to show the advantage of the new algorithm.The accuracy of the obtained numerical solution is compared with that of its original version(i.e.,the parareal algorithm based on the same numerical method).
基金supported by Science Challenge Project(Grant No.TZ2016002)the National Natural Science Foundation of China(Grant Nos.41874086 and 11971069)+1 种基金173 Program of China(Grant No.2020-JCJQ-ZD-029)the Excellent Youth Foundation of Hunan Province of China(Grant No.2018JJ1042).
文摘Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.
基金Kejia Pan was supported by the National Natural Science Foundation of China(Nos.41474103 and 41204082)the National High Technology Research and Development Program of China(No.2014AA06A602)+3 种基金the Natural Science Foundation of Hunan Province of China(No.2015JJ3148)Dongdong He was supported by the Fundamental Research Funds for the Central Universities,the National Natural Science Foundation of China(No.11402174)the Program for Young Excellent Talents at Tongji University(No.2013KJ012)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry。
文摘This paper proposes an extrapolation cascadic multigrid(EXCMG)method to solve elliptic problems in domains with reentrant corners.On a class ofλ-graded meshes,we derive some new extrapolation formulas to construct a high-order approximation to the finite element solution on the next finer mesh using the numerical solutions on two-level of grids(current and previous grids).Then,this high-order approximation is used as the initial guess to reduce computational cost of the conjugate gradient method.Recursive application of this idea results in the EXCMG method proposed in this paper.Finally,numerical results for a crack problem and an L-shaped problem are presented to verify the efficiency and effectiveness of the proposed EXCMG method.
文摘This paper deals with numerical computation and analysis for the initial boundary problems of two dimensional(2D)Sobolev equations with piecewise continuous argument.Firstly,a two-level high-order compact difference method(HOCDM)with computational accuracy O(τ^(2)+h_(x)^(4)+h_(y)^(4))is suggested,whereτ,h_(x),h_(y) denote the temporal and spatial stepsizes of the method,respectively.In order to improve the temporal computational accuracy of this method,the Richardson extrapolation technique is used and thus a new two-level HOCDMis derived,which is proved to be convergent of order four both in time and space.Although the new two-level HOCDM has the higher computational accuracy in time than the previous one,it will bring a larger computational cost.To overcome this deficiency,a three-level HOCDM with computational accuracy O(τ^(4)+h_(x)^(4)+h_(y)^(4))is constructed.Finally,with a series of numerical experiments,the theoretical accuracy and computational efficiency of the above methods are further verified.
基金This work is supported by the Foundation for Talent Introduction of Guangdong Provincial University, Guang- dong Province Universities and Colleges Pearl River Scholar Funded Scheme （2008）, and the National Natural Science Foundation of China under Grant No. 10971074.
文摘This paper applies a difference scheme to a singularly perturbed problem. The authors provide two algorithms on moving mesh methods by using Richardson extrapolation which can improve the accuracy of numerical solution. In traditional algorithms of moving meshes, the initial mesh is a uniform mesh. The authors change it to Bakhvalov-Shishkin mesh, and prove that it improves efficiency by numerical experiments. Finally, the results of the two algorithms are analyzed.
