As the generalization of the integer order partial differential equations(PDE),the fractional order PDEs are drawing more and more attention for their applications in fluid flow,finance and other areas.This paper pres...As the generalization of the integer order partial differential equations(PDE),the fractional order PDEs are drawing more and more attention for their applications in fluid flow,finance and other areas.This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin(DG)methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in space.The Caputo derivative is chosen as the representation of spatial derivative,because it may represent the fractional derivative by an integral operator.Some numerical examples show that the convergence orders of the proposed local Pk–DG methods are O(hk+1)both in one and two dimensions,where Pk denotes the space of the real-valued polynomials with degree at most k.展开更多
In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory(WENO)schemes to solve the one-dimensional and two-dimensional shallow...In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory(WENO)schemes to solve the one-dimensional and two-dimensional shallow water equations with source terms.Multilevel time discretization methods can make full use of computed information by WENO spatial discretization and save CPU cost by holding the former computational values.Extensive simulations are performed,which indicate that,the finite difference WENO schemes with multilevel time discretization can achieve higher accuracy,and are more cost effective than WENO scheme with Runge-Kutta time discretization,while still maintaining nonoscillatory properties.展开更多
In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Tota...In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws.The key idea of HWENO is to evolve both with the solution and its derivative,which allows for using Hermite interpolation in the reconstruction phase,resulting in a more compact stencil at the expense of the additional work.The main difference between this work and the formal one is the procedure to reconstruct the derivative terms.Comparing with the original HWENO schemes of Qiu and Shu,one major advantage of new HWENOschemes is its robust in computation of problem with strong shocks.Extensive numerical experiments are performed to illustrate the capability of the method.展开更多
基金supported by the National Natural Science Foundation of China for the Youth(No.10901157/A0117)the National Basic Research Program of China(973 Program 2012CB025904)+3 种基金supported by the National Basic Research Program under the Grant 2005CB321703the National Natural Science Foundation of China(No.10925101,10828101)the Program for New Century Excellent Talents in University(NCET-07-0022)the Doctoral Program of Education Ministry of China(No.20070001036).
文摘As the generalization of the integer order partial differential equations(PDE),the fractional order PDEs are drawing more and more attention for their applications in fluid flow,finance and other areas.This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin(DG)methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in space.The Caputo derivative is chosen as the representation of spatial derivative,because it may represent the fractional derivative by an integral operator.Some numerical examples show that the convergence orders of the proposed local Pk–DG methods are O(hk+1)both in one and two dimensions,where Pk denotes the space of the real-valued polynomials with degree at most k.
基金supported by NSFC 40906048NSFC 41040042+1 种基金NSFC 40801200Science research fund of Nanjing University of information science&technology 20090203.
文摘In this paper we study a class of multilevel high order time discretization procedures for the finite difference weighted essential non-oscillatory(WENO)schemes to solve the one-dimensional and two-dimensional shallow water equations with source terms.Multilevel time discretization methods can make full use of computed information by WENO spatial discretization and save CPU cost by holding the former computational values.Extensive simulations are performed,which indicate that,the finite difference WENO schemes with multilevel time discretization can achieve higher accuracy,and are more cost effective than WENO scheme with Runge-Kutta time discretization,while still maintaining nonoscillatory properties.
基金the National Natural Science Foundation of China(Grant No.10671097)the European project ADIGMA on the development of innovative solution algorithms for aerodynamic simu-lations+1 种基金Scientific Research Foundation for the Returned Overseas Chinese ScholarsState Education Ministry and the Natural Science Foundation of Jiangsu Province(Grant No.BK2006511)
文摘In this paper,we developed a class of the fourth order accurate finite volume Hermite weighted essentially non-oscillatory(HWENO)schemes based on the work(Computers&Fluids,34:642-663(2005))by Qiu and Shu,with Total Variation Diminishing Runge-Kutta time discretization method for the two-dimensional hyperbolic conservation laws.The key idea of HWENO is to evolve both with the solution and its derivative,which allows for using Hermite interpolation in the reconstruction phase,resulting in a more compact stencil at the expense of the additional work.The main difference between this work and the formal one is the procedure to reconstruct the derivative terms.Comparing with the original HWENO schemes of Qiu and Shu,one major advantage of new HWENOschemes is its robust in computation of problem with strong shocks.Extensive numerical experiments are performed to illustrate the capability of the method.
