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Ian Turner - One of the best experts on this subject based on the ideXlab platform.

  • a crank nicolson adi galerkin legendre spectral method for the two dimensional riesz space distributed order advection diffusion Equation
    Computers & Mathematics With Applications, 2018
    Co-Authors: Hui Zhang, Xiaoyun Jiang, Fanhai Zeng, Ian Turner
    Abstract:

    Abstract In the paper, a Crank–Nicolson alternating direction implicit (ADI) Galerkin–Legendre spectral scheme is presented for the two-dimensional Riesz space distributed-order advection–diffusion Equation. The Gauss quadrature has a higher computational accuracy than the mid-point quadrature rule, which is proposed to approximate the distributed order Riesz space derivative so that the considered Equation is transformed into a multi-term fractional Equation. Moreover, the transformed Equation is solved by discretizing in space by the ADI Galerkin–Legendre spectral scheme and in time using the Crank–Nicolson difference method. Stability and convergence analysis are verified for the numerical approximation. A lot of numerical results are demonstrated to justify the theoretical analysis.

  • a novel finite volume method for the riesz space distributed order advection diffusion Equation
    Applied Mathematical Modelling, 2017
    Co-Authors: Fawang Liu, Libo Feng, Ian Turner
    Abstract:

    Abstract In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) Equation. The mid-point quadrature rule is used to approximate the distributed-order Equation by a multi-term fractional model. Next, the transformed multi-term fractional Equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 γ

  • a novel finite volume method for the riesz space distributed order advection diffusion Equation
    ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS); Science & Engineering Faculty, 2017
    Co-Authors: Fawang Liu, Libo Feng, Ian Turner
    Abstract:

    In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) Equation. The mid-point quadrature rule is used to approximate the distributed-order Equation by a multi-term fractional model. Next, the transformed multi-term fractional Equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy. Finally, we give some examples to show the effectiveness of the numerical method. Keywords Distributed-order Equation; Finite volume method; Riesz fractional derivatives; Fractional advection–diffusion Equation; Stability and convergence

  • a characteristic difference method for the variable order fractional advection diffusion Equation
    Journal of Applied Mathematics and Computing, 2013
    Co-Authors: Shujun Shen, Ian Turner, Fawang Liu, Vo Anh, J Chen
    Abstract:

    In this paper, we consider a variable-order fractional Advection-Diffusion Equation with a nonlinear source term (VOFADE-NST) on a finite domain. Combining the characteristic method and the finite difference method, a characteristic finite difference method for solving the VOFADE-NST is presented. Its stability and convergence are analyzed. This new method is shown to be more efficient and superior to the standard finite difference method. Numerical experiments are carried out and the results demonstrate the effectiveness of theoretical analysis.

  • numerical simulation for the variable order galilei invariant advection diffusion Equation with a nonlinear source term
    Applied Mathematics and Computation, 2011
    Co-Authors: Changming Chen, Fawang Liu, Vo Anh, Ian Turner
    Abstract:

    In this paper, we consider the variable-order Galilei advection diffusion Equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the Equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis.

Rubayyi T Alqahtani - One of the best experts on this subject based on the ideXlab platform.

Thomas F. Russell - One of the best experts on this subject based on the ideXlab platform.

  • Eulerian-Lagrangian Localized Adjoint Methods for a Nonlinear Advection-Diffusion Equation
    Computer Methods in Applied Mechanics and Engineering, 1995
    Co-Authors: Helge K. Dahle, Richard E. Ewing, Thomas F. Russell
    Abstract:

    Eulerian-Lagrangian localized adjoint methods (ELLAM) are developed for the non-linear Buckley-Leverett Equation, which is characterized by degenerate diffusion and sharpening near-shock solutions. The ELLAM methodology employs space-time finite elements with edges oriented along flow paths, and space-time test functions that satisfy a local adjoint condition. This combination extends Eulerian-Lagrangian concepts in a systematic mass-conservative fashion to problems with general boundary conditions. Various kinds of boundary conditions are considered, and a local time-stepping procedure is developed for a no-flow outlet condition that leads to a boundary layer. Numerical experiments illustrate the potential of these methods.

Fawang Liu - One of the best experts on this subject based on the ideXlab platform.

  • a novel finite volume method for the riesz space distributed order advection diffusion Equation
    Applied Mathematical Modelling, 2017
    Co-Authors: Fawang Liu, Libo Feng, Ian Turner
    Abstract:

    Abstract In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) Equation. The mid-point quadrature rule is used to approximate the distributed-order Equation by a multi-term fractional model. Next, the transformed multi-term fractional Equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 γ

  • a novel finite volume method for the riesz space distributed order advection diffusion Equation
    ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS); Science & Engineering Faculty, 2017
    Co-Authors: Fawang Liu, Libo Feng, Ian Turner
    Abstract:

    In this paper, we investigate the finite volume method (FVM) for a distributed-order space-fractional advection–diffusion (AD) Equation. The mid-point quadrature rule is used to approximate the distributed-order Equation by a multi-term fractional model. Next, the transformed multi-term fractional Equation is solved by discretizing in space by the finite volume method and in time using the Crank–Nicolson scheme. We use a novel technique to deal with the convection term, by which the Riesz fractional derivative of order 0 < γ < 1 is transformed into a fractional integral form. An important contribution of our work is the use of nodal basis function to derive the discrete form of our model. The unique solvability of the scheme is also discussed and we prove that the Crank–Nicolson scheme is unconditionally stable and convergent with second-order accuracy. Finally, we give some examples to show the effectiveness of the numerical method. Keywords Distributed-order Equation; Finite volume method; Riesz fractional derivatives; Fractional advection–diffusion Equation; Stability and convergence

  • a characteristic difference method for the variable order fractional advection diffusion Equation
    Journal of Applied Mathematics and Computing, 2013
    Co-Authors: Shujun Shen, Ian Turner, Fawang Liu, Vo Anh, J Chen
    Abstract:

    In this paper, we consider a variable-order fractional Advection-Diffusion Equation with a nonlinear source term (VOFADE-NST) on a finite domain. Combining the characteristic method and the finite difference method, a characteristic finite difference method for solving the VOFADE-NST is presented. Its stability and convergence are analyzed. This new method is shown to be more efficient and superior to the standard finite difference method. Numerical experiments are carried out and the results demonstrate the effectiveness of theoretical analysis.

  • numerical approximations and solution techniques for the space time riesz caputo fractional advection diffusion Equation
    Numerical Algorithms, 2011
    Co-Authors: Shujun Shen, Fawang Liu, Vo Anh
    Abstract:

    In this paper, we consider a space-time Riesz---Caputo fractional Advection-Diffusion Equation. The Equation is obtained from the standard Advection-Diffusion Equation by replacing the first-order time derivative by the Caputo fractional derivative of order ????(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β 1???(0,1) and β 2???(1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the Equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.

  • numerical simulation for the variable order galilei invariant advection diffusion Equation with a nonlinear source term
    Applied Mathematics and Computation, 2011
    Co-Authors: Changming Chen, Fawang Liu, Vo Anh, Ian Turner
    Abstract:

    In this paper, we consider the variable-order Galilei advection diffusion Equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the Equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis.

P Ackerer - One of the best experts on this subject based on the ideXlab platform.

  • solving the advection diffusion Equation with the eulerian lagrangian localized adjoint method on unstructured meshes and non uniform time stepping
    Journal of Computational Physics, 2005
    Co-Authors: Anis Younes, P Ackerer
    Abstract:

    Eulerian-Lagrangian localized adjoint method (ELLAM) is used to solve the advection diffusion Equation (ADE) which is a very common mathematical model in physics. In this work, ELLAM is extended to triangular meshes. Standard integration schemes, which perform well for rectangular grids, are improved to reduce oscillations with unstructured triangulations. Numerical experiments for grid Peclet numbers ranking from 1 to 100 show the efficiency of the developed scheme. A new algorithm is also developed in order to avoid excessive numerical diffusion when using many time steps with the ELLAM. The basic idea of this approach is to keep the same characteristics for all time steps and to interpolate only the concentration variations due to the dispersion process at the end of each time step. Although ELLAM requires a lot of integration points for unstructured meshes, it remains a competitive method when using a single or many time steps compared to explicit discontinuous Galerkin finite element method.