Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these m...Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.展开更多
In this paper,strategies are provided for a powerful numerical manifold method(NMM)with h and p refinement in analyses of elasticity and elasto-plasticity.The new NMM is based on the concept of the independent cover,w...In this paper,strategies are provided for a powerful numerical manifold method(NMM)with h and p refinement in analyses of elasticity and elasto-plasticity.The new NMM is based on the concept of the independent cover,which gets rid of NMM's important defect of rank deficiency when using higher-order local approximation functions.Several techniques are presented.In terms of mesh generation,a relationship between the quadtree structure and the mathematical mesh is established to allow a robust h-refinement.As to the condition number,a scaling based on the physical patch is much better than the classical scaling based on the mathematical patch;an overlapping width of 1%–10%can ensure a good condition number for 2nd,3rd,and 4th order local approximation functions;the small element issue can be overcome after the local approximation on small patch is replaced by that on a regular patch.On numerical accuracy,local approximation using complete polynomials is necessary for the optimal convergence rate.Two issues that may damage the convergence rate should be prevented.The first is to approximate the curved boundary of a higher-order element by overly few straight lines,and the second is excessive overlapping width.Finally,several refinement strategies are verified by numerical examples.展开更多
The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex...The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex variable are non-integers.This kind of systems can be approximated by high-order integer-order systems,and can be analyzed and designed by the sophisticated integer-order systems methodology.A new closed-form algorithm for fractional-order linear differential equations is proposed based on the definitions of fractional-order derivatives,and the effectiveness of the algorithm is illustrated through examples.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
For a general nonlinear fractional-order differential equation, the numerical solution is a good way to approximate the trajectory of such systems. In this paper, a novel algorithm for numerical solution of fractional...For a general nonlinear fractional-order differential equation, the numerical solution is a good way to approximate the trajectory of such systems. In this paper, a novel algorithm for numerical solution of fractional-order differential equations based on the definition of Grunwald-Letnikov is presented. The results of numerical solution by using the novel method and the frequency-domain method are compared, and the limitations of frequency-domain method are discussed.展开更多
基金The research of the first author is support by NSFC grant 10601055,FANEDD of CAS and SRF for ROCS SEMThe research of the second author is supported by NSF grant DMS-0809086 and DOE grant DE-FG02-08ER25863.
文摘Discontinuous Galerkin (DG) methods are a class of finite element methodsusing discontinuous basis functions, which are usually chosen as piecewise polynomi-als. Since the basis functions can be discontinuous, these methods have the flexibilitywhich is not shared by typical finite element methods, such as the allowance of ar-bitrary triangulation with hanging nodes, less restriction in changing the polynomialdegrees in each element independent of that in the neighbors (p adaptivity), and localdata structure and the resulting high parallel efficiency. In this paper, we give a generalreview of the local DG (LDG) methods for solving high-order time-dependent partialdifferential equations (PDEs). The important ingredient of the design of LDG schemes,namely the adequate choice of numerical fluxes, is highlighted. Some of the applica-tions of the LDG methods for high-order time-dependent PDEs are also be discussed.
基金supported by the National Natural Science Foundation of China(Grant Nos.52130905 and 52079002)。
文摘In this paper,strategies are provided for a powerful numerical manifold method(NMM)with h and p refinement in analyses of elasticity and elasto-plasticity.The new NMM is based on the concept of the independent cover,which gets rid of NMM's important defect of rank deficiency when using higher-order local approximation functions.Several techniques are presented.In terms of mesh generation,a relationship between the quadtree structure and the mathematical mesh is established to allow a robust h-refinement.As to the condition number,a scaling based on the physical patch is much better than the classical scaling based on the mathematical patch;an overlapping width of 1%–10%can ensure a good condition number for 2nd,3rd,and 4th order local approximation functions;the small element issue can be overcome after the local approximation on small patch is replaced by that on a regular patch.On numerical accuracy,local approximation using complete polynomials is necessary for the optimal convergence rate.Two issues that may damage the convergence rate should be prevented.The first is to approximate the curved boundary of a higher-order element by overly few straight lines,and the second is excessive overlapping width.Finally,several refinement strategies are verified by numerical examples.
基金supported by the National Natural Science Foundation of China (Grant No.60475036).
文摘The definitions and properties of widely used fractional-order derivatives are summarized in this paper.The characteristic polynomials of the fractional-order systems are pseudo-polynomials whose powers of the complex variable are non-integers.This kind of systems can be approximated by high-order integer-order systems,and can be analyzed and designed by the sophisticated integer-order systems methodology.A new closed-form algorithm for fractional-order linear differential equations is proposed based on the definitions of fractional-order derivatives,and the effectiveness of the algorithm is illustrated through examples.
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.
基金the Natural Science Foundation of CQ CSTC under Grant No. 2007BB2161.
文摘For a general nonlinear fractional-order differential equation, the numerical solution is a good way to approximate the trajectory of such systems. In this paper, a novel algorithm for numerical solution of fractional-order differential equations based on the definition of Grunwald-Letnikov is presented. The results of numerical solution by using the novel method and the frequency-domain method are compared, and the limitations of frequency-domain method are discussed.
