In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.T...In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.展开更多
In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the origi...In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.展开更多
长距离输水管道水力瞬变过程中水体压强达到汽化压强时,将会发生水柱分离现象,水柱弥合将产生异常高压,导致管路振动、变形甚至爆管事故。已有的水柱分离弥合水锤数学模型主要采用特征线法(Method of characteristics,MOC)计算,并且很...长距离输水管道水力瞬变过程中水体压强达到汽化压强时,将会发生水柱分离现象,水柱弥合将产生异常高压,导致管路振动、变形甚至爆管事故。已有的水柱分离弥合水锤数学模型主要采用特征线法(Method of characteristics,MOC)计算,并且很少考虑动态摩阻引起的能量衰减。为提高水柱分离弥合水锤现象的计算精确度和稳定性,基于有限体积法二阶Godunov格式,建立了考虑动态摩阻的离散气体空穴模型(Discrete gas cavity model,DGCM)。为实现管道边界和内部单元的统一计算,提出虚拟边界的处理方法。将该模型模拟结果与实验数据以及已有的稳态摩阻模型的计算结果进行比较,并对网格数、压力修正系数等参数敏感性进行分析。结果表明,本模型能够准确模拟出纯水锤、水柱分离弥合水锤两种情况下瞬变压力,与实验数据基本一致;考虑动态摩阻的瞬态压力计算值与实验数据更吻合;与MOC相比,当库朗数小于1.0时,有限体积法二阶Godunov模型计算结果更准确、更稳定;尤其是,压力修正系数取值0.9及较密网格时数学模型能更为准确地再现实验结果。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
In this paper,we study a second-order accurate and linear numerical scheme for the nonlocal CahnHilliard equation.The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth ...In this paper,we study a second-order accurate and linear numerical scheme for the nonlocal CahnHilliard equation.The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization,and by applying the Fourier spectral collocation to the spatial discretization.In addition,two stabilization terms in different forms are added for the sake of the numerical stability.We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme,combined with the rough error estimate and the refined estimate.By regarding the numerical solution as a small perturbation of the exact solution,we are able to justify the discrete?^(∞)bound of the numerical solution,as a result of the rough error estimate.Subsequently,the refined error estimate is derived to obtain the optimal rate of convergence,following the established?∞bound of the numerical solution.Moreover,the energy stability is also rigorously proved with respect to a modified energy.The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work,and the energy stability estimate has greatly improved the corresponding result therein.展开更多
We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, ...We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.展开更多
The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with sma...The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.展开更多
基金This work is supported by the National Natural Science Foundation of China(11661058,11761053)the Natural Science Foundation of Inner Mongolia(2017MS0107)the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region(NJYT-17-A07).
文摘In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.
文摘In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.
文摘长距离输水管道水力瞬变过程中水体压强达到汽化压强时,将会发生水柱分离现象,水柱弥合将产生异常高压,导致管路振动、变形甚至爆管事故。已有的水柱分离弥合水锤数学模型主要采用特征线法(Method of characteristics,MOC)计算,并且很少考虑动态摩阻引起的能量衰减。为提高水柱分离弥合水锤现象的计算精确度和稳定性,基于有限体积法二阶Godunov格式,建立了考虑动态摩阻的离散气体空穴模型(Discrete gas cavity model,DGCM)。为实现管道边界和内部单元的统一计算,提出虚拟边界的处理方法。将该模型模拟结果与实验数据以及已有的稳态摩阻模型的计算结果进行比较,并对网格数、压力修正系数等参数敏感性进行分析。结果表明,本模型能够准确模拟出纯水锤、水柱分离弥合水锤两种情况下瞬变压力,与实验数据基本一致;考虑动态摩阻的瞬态压力计算值与实验数据更吻合;与MOC相比,当库朗数小于1.0时,有限体积法二阶Godunov模型计算结果更准确、更稳定;尤其是,压力修正系数取值0.9及较密网格时数学模型能更为准确地再现实验结果。
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 Chinese Academy of Sciences(CAS)Academy of Mathematics and Systems Science(AMSS)the Hong Kong Polytechnic University(PolyU)Joint Laboratory of Applied Mathematics+4 种基金supported by the Hong Kong Research Council General Research Fund(Grant No.15300821)the Hong Kong Polytechnic University Grants(Grant Nos.1-BD8N,4-ZZMK and 1-ZVWW)supported by the Hong Kong Research Council Research Fellow Scheme(Grant No.RFS2021-5S03)General Research Fund(Grant No.15302919)supported by US National Science Foundation(Grant No.DMS-2012269)。
文摘In this paper,we study a second-order accurate and linear numerical scheme for the nonlocal CahnHilliard equation.The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization,and by applying the Fourier spectral collocation to the spatial discretization.In addition,two stabilization terms in different forms are added for the sake of the numerical stability.We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme,combined with the rough error estimate and the refined estimate.By regarding the numerical solution as a small perturbation of the exact solution,we are able to justify the discrete?^(∞)bound of the numerical solution,as a result of the rough error estimate.Subsequently,the refined error estimate is derived to obtain the optimal rate of convergence,following the established?∞bound of the numerical solution.Moreover,the energy stability is also rigorously proved with respect to a modified energy.The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work,and the energy stability estimate has greatly improved the corresponding result therein.
基金Project supported by the National Natural Science Foundation of China(Grant No.11471262)the National Basic Research Program of China(Grant No.2012CB025904)the State Key Laboratory of Science and Engineering Computing and the Center for High Performance Computing of Northwestern Polytechnical University,China
文摘We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.
基金Project supported by National Natural Science Foundation of China.
文摘The bounds on the discrepancy of approximate solutions constructed by Gedunov's scheme to IVP of isentropic equations of gas dynamics are obtained, Three well-knowu results obtained by Lax for shock waves with small jumps for general quasilinear hyperbolic systems of conservation laws are extended to shock waves for isentropic equations of gas dynamics in a bounded invariant region with ρ=0 as one of boundries of the region. Two counterexamples are given to show that two iuequalities given by Godunov do not hold for all rational numbers γ∈(1, 3]. It seems that the approach by Godunov to obtain the forementioned bounds may not be possible.
