Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing ...Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.展开更多
In this paper, we propose an improved preconditioned algorithm for the conjugate gradient squared method (improved PCGS) for the solution of linear equations. Further, the logical structures underlying the formation o...In this paper, we propose an improved preconditioned algorithm for the conjugate gradient squared method (improved PCGS) for the solution of linear equations. Further, the logical structures underlying the formation of this preconditioned algorithm are demonstrated via a number of theorems. This improved PCGS algorithm retains some mathematical properties that are associated with the CGS derivation from the bi-conjugate gradient method under a non-preconditioned system. A series of numerical comparisons with the conventional PCGS illustrate the enhanced effectiveness of our improved scheme with a variety of preconditioners. This logical structure underlying the formation of the improved PCGS brings a spillover effect from various bi-Lanczos-type algorithms with minimal residual operations, because these algorithms were constructed by adopting the idea behind the derivation of CGS. These bi-Lanczos-type algorithms are very important because they are often adopted to solve the systems of linear equations that arise from large-scale numerical simulations.展开更多
This paper deals with numerical methods for solving one-dimensional(1D)and twodimensional(2D)initial-boundary value problems(IBVPs)of space-fractional sine-Gordon equations(SGEs)with distributed delay.For 1D problems,...This paper deals with numerical methods for solving one-dimensional(1D)and twodimensional(2D)initial-boundary value problems(IBVPs)of space-fractional sine-Gordon equations(SGEs)with distributed delay.For 1D problems,we construct a kind of oneparameter finite difference(OPFD)method.It is shown that,under a suitable condition,the proposed method is convergent with second order accuracy both in time and space.In implementation,the preconditioned conjugate gradient(PCG)method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method.For 2D problems,we develop another kind of OPFD method.For such a method,two classes of accelerated schemes are suggested,one is alternative direction implicit(ADI)scheme and the other is ADI-PCG scheme.In particular,we prove that ADI scheme can arrive at second-order accuracy in time and space.With some numerical experiments,the computational effectiveness and accuracy of the methods are further verified.Moreover,for the suggested methods,a numerical comparison in computational efficiency is presented.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.5130926141030747+3 种基金41102181and 51121005)the National Basic Research Program of China(973 Program)(No.2011CB013503)the Young Teachers’ Initial Funding Scheme of Sun Yat-sen University(No.39000-1188140)
文摘Fast solving large-scale linear equations in the finite element analysis is a classical subject in computational mechanics. It is a key technique in computer aided engineering （CAE） and computer aided manufacturing （CAM）. This paper presents a high-efficiency improved symmetric successive over-relaxation （ISSOR） preconditioned conjugate gradient （PCG） method, which maintains lelism consistent with the original form. Ideally, the by 50% as compared with the original algorithm. the convergence and inherent paralcomputation can It is suitable for be reduced nearly high-performance computing with its inherent basic high-efficiency operations. By comparing with the numerical results, it is shown that the proposed method has the best performance.
文摘In this paper, we propose an improved preconditioned algorithm for the conjugate gradient squared method (improved PCGS) for the solution of linear equations. Further, the logical structures underlying the formation of this preconditioned algorithm are demonstrated via a number of theorems. This improved PCGS algorithm retains some mathematical properties that are associated with the CGS derivation from the bi-conjugate gradient method under a non-preconditioned system. A series of numerical comparisons with the conventional PCGS illustrate the enhanced effectiveness of our improved scheme with a variety of preconditioners. This logical structure underlying the formation of the improved PCGS brings a spillover effect from various bi-Lanczos-type algorithms with minimal residual operations, because these algorithms were constructed by adopting the idea behind the derivation of CGS. These bi-Lanczos-type algorithms are very important because they are often adopted to solve the systems of linear equations that arise from large-scale numerical simulations.
基金supported by the NSFC(Grant No.11971010)the Science and Technology Development Fund of Macao(Grant No.0122/2020/A3)MYRG2020-00224-FST from University of Macao,China.
文摘This paper deals with numerical methods for solving one-dimensional(1D)and twodimensional(2D)initial-boundary value problems(IBVPs)of space-fractional sine-Gordon equations(SGEs)with distributed delay.For 1D problems,we construct a kind of oneparameter finite difference(OPFD)method.It is shown that,under a suitable condition,the proposed method is convergent with second order accuracy both in time and space.In implementation,the preconditioned conjugate gradient(PCG)method with the Strang circulant preconditioner is carried out to improve the computational efficiency of the OPFD method.For 2D problems,we develop another kind of OPFD method.For such a method,two classes of accelerated schemes are suggested,one is alternative direction implicit(ADI)scheme and the other is ADI-PCG scheme.In particular,we prove that ADI scheme can arrive at second-order accuracy in time and space.With some numerical experiments,the computational effectiveness and accuracy of the methods are further verified.Moreover,for the suggested methods,a numerical comparison in computational efficiency is presented.
基金The work was supported in part by Natural Science Foundation of China(10372087) by Natural Science Foundation of China and Associated Foundation of China Academy of Engineering Physics(10376031)
