The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are est...The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.展开更多
In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-depe...In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.展开更多
We describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extension...We describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extensions to equations with more general (nonlinear) vanishing delays are also discussed.展开更多
The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying...The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.The numerical examples are given to illustrate the theoretical results.展开更多
In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then...In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.展开更多
Diagonalized Chebyshev rational spectral methods for solving second-order elliptic problems on the half/whole line are proposed.Some Sobolev bi-orthogonal rational basis functions are constructed which lead to the dia...Diagonalized Chebyshev rational spectral methods for solving second-order elliptic problems on the half/whole line are proposed.Some Sobolev bi-orthogonal rational basis functions are constructed which lead to the diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like Chebyshev rational series.Numerical results demonstrate the effectiveness of the suggested approaches.展开更多
The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc,using Fourier and generalized Laguerre functions.As an example,we consider a nonlinear strongly d...The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc,using Fourier and generalized Laguerre functions.As an example,we consider a nonlinear strongly damped wave equation.The mixed spectral and pseudospectral schemes are proposed.The convergence is proved.Numerical results demonstrate the efficiency of this approach.展开更多
We develop and analyze a first-order system least-squares spectral method for the second-order elhptic boundary value problem with variable coefficients. We first analyze the Chebyshev weighted norm least-squares func...We develop and analyze a first-order system least-squares spectral method for the second-order elhptic boundary value problem with variable coefficients. We first analyze the Chebyshev weighted norm least-squares functional defined by the sum of the Lw^2- and Hw^-1- norm of the residual equations and then we eplace the negative norm by the discrete negative norm and analyze the discrete Chebyshev weighted least-squares method. The spectral convergence is derived for the proposed method. We also present various numerical experiments. The Legendre weighted least-squares method can be easily developed by following this paper.展开更多
Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data,the initial conditions and the operator coe...Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data,the initial conditions and the operator coefficients.Sharp/strict maximum principles insomuch of fundamental importance for the continuous problem often do not persist under numerical discretization.A lot of past research concentrates on designing fine numerical schemes which preserves the sharp maximum principles especially for nonlinear problems.However these sharp principles not only sometimes introduce unwanted stringent conditions on the numerical schemes but also completely leaves many powerful frequency-based methods unattended and rarely analyzed directly in the sharp ma-ximum norm topology.A prominent example is the spectral methods in the family of weighted residual methods.In this work we introduce and develop a new framework of almost sharp maximum principles which allow the numerical solutions to deviate from the sharp bound by a controllable discretization error:we call them effective maximum principles.We showcase the analysis for the classical Fourier spectral methods including Fourier Galerkin and Fourier collocation in space with forward Euler in time or second order Strang splitting.The model equations include the Allen-Cahn equations with double well potential,the Burgers equation and the Navier-Stokes equations.We give a comprehensive proof of the effective maximum principles under very general parametric conditions.展开更多
In this paper,efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave.By using the Caputo fractional derivative definition...In this paper,efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave.By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator,finite difference method in time and spectral method in space are constructed for the considered model.The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term.The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time,and spectral accuracy in space.Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims.Finally,the decay rate of solutions are investigated.展开更多
In this paper,we introduce the dissipative spectral methods(DSM)for the first order linear hyperbolic equations in one dimension.Specifically,we consider the Fourier DSM for periodic problems and the Legendre DSM for ...In this paper,we introduce the dissipative spectral methods(DSM)for the first order linear hyperbolic equations in one dimension.Specifically,we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition.The error estimates of the methods are shown to be quasioptimal for variable-coefficients equations.Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods,showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.展开更多
A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the c...A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.展开更多
In this study,application of the spectral representation method for generation of endurance time excitation functions is introduced.Using this method,the intensifying acceleration time series is generated so that its ...In this study,application of the spectral representation method for generation of endurance time excitation functions is introduced.Using this method,the intensifying acceleration time series is generated so that its acceleration response spectrum in any desired time duration is compatible with a time-scaled predefined acceleration response spectrum.For this purpose,simulated stationary acceleration time series is multiplied by the time dependent linear modulation function,then using a simple iterative scheme,it is forced to match a target acceleration response spectrum.It is shown that the generated samples have excellent conformity in low frequency,which is useful for nonlinear endurance time analysis.In the second part of this study,it is shown that this procedure can be extended to generate a set of spatially correlated endurance time excitation functions.This makes it possible to assess the performance of long structures under multi-support seismic excitation using endurance time analysis.展开更多
In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy o...In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.展开更多
The spectral methods and ice-induced fatigue analysis are discussed based on Miner's linear cumulative fatigue hypothesis and S-N curve data. According to the long-term data of full-scale tests on the platforms in th...The spectral methods and ice-induced fatigue analysis are discussed based on Miner's linear cumulative fatigue hypothesis and S-N curve data. According to the long-term data of full-scale tests on the platforms in the Bohai Sea, the ice force spectrum of conical structures and the fatigue environmental model are established. Moreover, the finite element model of JZ20-2MSW platform, an example of ice-induced fatigue analysis, is built with ANSYS software. The mode analysis and dynamic analysis in frequency domain under all kinds of ice fatigue work conditions are carded on, and the fatigue life of the structure is estimated in detail. The methods in this paper can be helpful in ice-induced fatigue analysis of ice-resistant platforms.展开更多
The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution ...The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d(AN,A) → 0 are proved.展开更多
This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating ...This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.展开更多
In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time ...In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.展开更多
Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple the...Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.展开更多
In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transforma...In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.展开更多
基金the Science Foundation of the Science and Technology Commission of Shanghai Municipality(No.075105118)the Shanghai Leading Academic Discipline Project(No.T0401)the Fund for E-institute of Shanghai Universities(No.E03004)
文摘The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.
文摘In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.
文摘We describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown that the resulting numerical solutions exhibit the spectral convergence order. Extensions to equations with more general (nonlinear) vanishing delays are also discussed.
基金This work is supported by the Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)National Science Foundation of China(10971074).
文摘The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel(t−s)^(−μ) with 0<μ<1.In this work,we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth.We provide a rigorous error analysis for the spectral methods,which shows that both the errors of approximate solutions and the errors of approximate derivatives of the solutions decay exponentially in L^(∞)-norm and weighted L^(2)-norm.The numerical examples are given to illustrate the theoretical results.
基金supported by the State Key Program of National Natural Science Foundation of China(No.11931003)National Natural Science Foundation of China(Nos.41974133,11671157 and 11971410)supported by the Innovation Project of Graduate School of South China Normal University(No.2018LKXM008).
文摘In this paper,an initial boundary value problem of the space-time fractional diffusion equation is studied.Both temporal and spatial directions for this equation are discreted by the Galerkin spectral methods.And then based on the discretization scheme,reliable a posteriori error estimates for the spectral approximation are derived.Some numerical examples are presented to verify the validity and applicability of the derived a posteriori error estimator.
基金This work was supported in part by National Natural Science Foun-dation of China(Nos.11571238 and 11601332).
文摘Diagonalized Chebyshev rational spectral methods for solving second-order elliptic problems on the half/whole line are proposed.Some Sobolev bi-orthogonal rational basis functions are constructed which lead to the diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like Chebyshev rational series.Numerical results demonstrate the effectiveness of the suggested approaches.
基金supported in part by NSF of China,N.10771142the National Basic Research Project of China,N.2005CB321701+2 种基金Shuguang Project of Shanghai Education Commission,N.08SG45Shanghai Leading Academic Discipline Project N.S30405The Fund for E-institute of Shanghai Universities N.E03004.
文摘The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc,using Fourier and generalized Laguerre functions.As an example,we consider a nonlinear strongly damped wave equation.The mixed spectral and pseudospectral schemes are proposed.The convergence is proved.Numerical results demonstrate the efficiency of this approach.
文摘We develop and analyze a first-order system least-squares spectral method for the second-order elhptic boundary value problem with variable coefficients. We first analyze the Chebyshev weighted norm least-squares functional defined by the sum of the Lw^2- and Hw^-1- norm of the residual equations and then we eplace the negative norm by the discrete negative norm and analyze the discrete Chebyshev weighted least-squares method. The spectral convergence is derived for the proposed method. We also present various numerical experiments. The Legendre weighted least-squares method can be easily developed by following this paper.
文摘Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data,the initial conditions and the operator coefficients.Sharp/strict maximum principles insomuch of fundamental importance for the continuous problem often do not persist under numerical discretization.A lot of past research concentrates on designing fine numerical schemes which preserves the sharp maximum principles especially for nonlinear problems.However these sharp principles not only sometimes introduce unwanted stringent conditions on the numerical schemes but also completely leaves many powerful frequency-based methods unattended and rarely analyzed directly in the sharp ma-ximum norm topology.A prominent example is the spectral methods in the family of weighted residual methods.In this work we introduce and develop a new framework of almost sharp maximum principles which allow the numerical solutions to deviate from the sharp bound by a controllable discretization error:we call them effective maximum principles.We showcase the analysis for the classical Fourier spectral methods including Fourier Galerkin and Fourier collocation in space with forward Euler in time or second order Strang splitting.The model equations include the Allen-Cahn equations with double well potential,the Burgers equation and the Navier-Stokes equations.We give a comprehensive proof of the effective maximum principles under very general parametric conditions.
基金The research of this author is partially supported by NSF of China(51661135011 and 91630204).
文摘In this paper,efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave.By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator,finite difference method in time and spectral method in space are constructed for the considered model.The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term.The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time,and spectral accuracy in space.Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims.Finally,the decay rate of solutions are investigated.
基金supported by National Natural Science Foundation of China(11171209)Leading Academic Discipline Project of Shanghai Municipal Education Commission(J50101)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(20060280010)Graduate Innovative Foundation of Shanghai University(SHUCX091048).
文摘In this paper,we introduce the dissipative spectral methods(DSM)for the first order linear hyperbolic equations in one dimension.Specifically,we consider the Fourier DSM for periodic problems and the Legendre DSM for equations with the Dirichlet boundary condition.The error estimates of the methods are shown to be quasioptimal for variable-coefficients equations.Numerical results are given to verify high accuracy of the DSM and to compare the proposed schemes with some high performance methods,showing some superiority in long-term integration for the periodic case and in dealing with limited smoothness near or at the boundary for the Dirichlet case.
基金the National Natural Science Foundation of China(No.11701103)the Young Top-notch Talent Program of Guangdong Province of China(No.2017GC010379)+4 种基金the Natural Science Foundation of Guangdong Province of China(No.2022A1515012147)the Project of Science and Technology of Guangzhou of China(No.202102020704)the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University of China(2021023)the Science and Technology Development Fund,Macao SAR(File No.0005/2019/A)the University of Macao of China(File Nos.MYRG2020-00035-FST,MYRG2018-00047-FST).
文摘A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.
文摘In this study,application of the spectral representation method for generation of endurance time excitation functions is introduced.Using this method,the intensifying acceleration time series is generated so that its acceleration response spectrum in any desired time duration is compatible with a time-scaled predefined acceleration response spectrum.For this purpose,simulated stationary acceleration time series is multiplied by the time dependent linear modulation function,then using a simple iterative scheme,it is forced to match a target acceleration response spectrum.It is shown that the generated samples have excellent conformity in low frequency,which is useful for nonlinear endurance time analysis.In the second part of this study,it is shown that this procedure can be extended to generate a set of spatially correlated endurance time excitation functions.This makes it possible to assess the performance of long structures under multi-support seismic excitation using endurance time analysis.
文摘In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.
基金The paper was supported by the National 863 High Technology Develpoment Plan Project(Grant No.2001AA602015)
文摘The spectral methods and ice-induced fatigue analysis are discussed based on Miner's linear cumulative fatigue hypothesis and S-N curve data. According to the long-term data of full-scale tests on the platforms in the Bohai Sea, the ice force spectrum of conical structures and the fatigue environmental model are established. Moreover, the finite element model of JZ20-2MSW platform, an example of ice-induced fatigue analysis, is built with ANSYS software. The mode analysis and dynamic analysis in frequency domain under all kinds of ice fatigue work conditions are carded on, and the fatigue life of the structure is estimated in detail. The methods in this paper can be helpful in ice-induced fatigue analysis of ice-resistant platforms.
基金This work was supported by the National Science Foundation of China(10271034)
文摘The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estimation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d(AN,A) → 0 are proved.
文摘This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method.A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established.This formula is expressed in terms of a certain terminating hypergeometric function of the type_(4)F_(3)(1).This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type 3 F 2(1)which can be summed with the aid of Watson’s identity.Six illustrative examples are presented to ensure the applicability and accuracy of the proposed algorithm.
基金supported by NSFC(Grant No.11601490).Research of Y.Xu is supported by NSFC(Grant No.12071455).
文摘In[20],a semi-implicit spectral deferred correction(SDC)method was proposed,which is efficient for highly nonlinear partial differential equations(PDEs).The semi-implicit SDC method in[20]is based on first-order time integration methods,which are corrected iteratively,with the order of accuracy increased by one for each additional iteration.In this paper,we will develop a class of semi-implicit SDC methods,which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration.For spatial discretization,we employ the local discontinuous Galerkin(LDG)method to arrive at fully-discrete schemes,which are high-order accurate in both space and time.Numerical experiments are presented to demonstrate the accuracy,efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.
基金supported by the NSFC Grant no.12271492the Natural Science Foundation of Henan Province of China Grant no.222300420550+1 种基金supported by the NSFC Grant no.12271498the National Key R&D Program of China Grant no.2022YFA1005202/2022YFA1005200.
文摘Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows,it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations.One popular and efficient strategy is to add an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them.The resulting schemes are only first-order accurate in time,and it seems extremely difficult to generalize the idea of stabilization to the second-order or higher version.In this paper,we employ the spectral deferred correction method to improve the temporal accuracy,based on the first-order decoupled and energy-stable scheme constructed by the stabilization idea.The novelty lies in how the decoupling and linear implicit properties are maintained to improve the efficiency.Within the framework of the spatially discretized local discontinuous Galerkin method,the resulting numerical schemes are fully decoupled,efficient,and high-order accurate in both time and space.Numerical experiments are performed to validate the high-order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.
基金supported by the State Key Program of National Natural Science Foundation of China(11931003)the National Natural Science Foundation of China(41974133,11671157)。
文摘In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.