7,093 research outputs found
Implicit finite difference solution for time-fractional diffusion equations using AOR method
In this paper, we derive an implicit finite difference approximation equation of the one-dimensional linear time fractional diffusion equations, based on the Caputo's time fractional derivative. Then this approximation equation leads the corresponding system of linear equation, which is large scale and sparse. Due to the characteristics of the coefficient matrix, we use the Accelerated Over-Relaxation (AOR) iterative method for solving the generated linear system. One example of the problem is presented to illustrate the effectiveness of AOR method. The numerical results of this study show that the proposed iterative method is superior compared with the existing one weighted parameter iterative method
The Innovation Iterative Method and its Stability in Time-Fractional Diffusion Equations
In this research, we deal with the innovation or application iterative methods of an unconditionally implicit finite difference approximation equation and the one-dimensional, linear time fractional diffusion equations (TFDEs) via Caputo’s time fractional derivative. Based on this implicit approximation equation, the corresponding linear system can be generated, in which its coefficient matrix is large scale and sparse. To speed up the convergence rate in solving the linear system iteratively, we construct the corresponding preconditioned linear system. Then we formulate and implement the Preconditioned Gauss-Seidel (PGS) iterative method for solving the generated linear system. Two examples of the problem are presented to illustrate the effectiveness of the PGS method. The two numerical results of this study show that the proposed iterative method is superior to the basic GS iterative method
Numerical solution of time-dependent problems with fractional power elliptic operator
An unsteady problem is considered for a space-fractional equation in a
bounded domain. A first-order evolutionary equation involves a fractional power
of an elliptic operator of second order. Finite element approximation in space
is employed. To construct approximation in time, standard two-level schemes are
used. The approximate solution at a new time-level is obtained as a solution of
a discrete problem with the fractional power of the elliptic operator. A
Pade-type approximation is constructed on the basis of special quadrature
formulas for an integral representation of the fractional power elliptic
operator using explicit schemes. A similar approach is applied in the numerical
implementation of implicit schemes. The results of numerical experiments are
presented for a test two-dimensional problem.Comment: 25 pages, 13 figures, 6 tables. arXiv admin note: text overlap with
arXiv:1510.08297, arXiv:1412.570
Numerical Approximations for Fractional Differential Equations
The Gr\"unwald and shifted Gr\"unwald formulas for the function
are first order approximations for the Caputo fractional derivative of the
function with lower limit at the point . We obtain second and third
order approximations for the Gr\"unwald and shifted Gr\"unwald formulas with
weighted averages of Caputo derivatives when sufficient number of derivatives
of the function are equal to zero at , using the estimate for the
error of the shifted Gr\"unwald formulas. We use the approximations to
determine implicit difference approximations for the sub-diffusion equation
which have second order accuracy with respect to the space and time variables,
and second and third order numerical approximations for ordinary fractional
differential equations
Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems
We study the hybridizable discontinuous Galerkin (HDG) method for the spatial
discretization of time fractional diffusion models with Caputo derivative of
order . For each time , the HDG approximations are
taken to be piecewise polynomials of degree on the spatial
domain~, the approximations to the exact solution in the
-norm and to in the
-norm are proven to converge with
the rate provided that is sufficiently regular, where is the
maximum diameter of the elements of the mesh. Moreover, for , we obtain
a superconvergence result which allows us to compute, in an elementwise manner,
a new approximation for converging with a rate (ignoring the
logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments
validating the theoretical results are displayed
A discontinuous Petrov-Galerkin method for time-fractional diffusion equations
We propose and analyze a time-stepping discontinuous Petrov-Galerkin method
combined with the continuous conforming finite element method in space for the
numerical solution of time-fractional subdiffusion problems. We prove the
existence, uniqueness and stability of approximate solutions, and derive error
estimates. To achieve high order convergence rates from the time
discretizations, the time mesh is graded appropriately near~ to compensate
the singular (temporal) behaviour of the exact solution near caused by
the weakly singular kernel, but the spatial mesh is quasiuniform. In the
-norm ( is the time domain and is
the spatial domain), for sufficiently graded time meshes, a global convergence
of order is shown, where is the
fractional exponent, is the maximum time step, is the maximum diameter
of the spatial finite elements, and and are the degrees of approximate
solutions in time and spatial variables, respectively. Numerical experiments
indicate that our theoretical error bound is pessimistic. We observe that the
error is of order ~, that is, optimal in both variables.Comment: SIAM Journal on Numerical Analysis, 201
Numerical solving unsteady space-fractional problems with the square root of an elliptic operator
An unsteady problem is considered for a space-fractional equation in a
bounded domain. A first-order evolutionary equation involves the square root of
an elliptic operator of second order. Finite element approximation in space is
employed. To construct approximation in time, regularized two-level schemes are
used. The numerical implementation is based on solving the equation with the
square root of the elliptic operator using an auxiliary Cauchy problem for a
pseudo-parabolic equation. The scheme of the second-order accuracy in time is
based on a regularization of the three-level explicit Adams scheme. More
general problems for the equation with convective terms are considered, too.
The results of numerical experiments are presented for a model two-dimensional
problem.Comment: 21 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1412.570
A scale-dependent finite difference method for time fractional derivative relaxation type equations
Fractional derivative relaxation type equations (FREs) including fractional
diffusion equation and fractional relaxation equation, have been widely used to
describe anomalous phenomena in physics. To utilize the characteristics of
fractional dynamic systems, this paper proposes a scale-dependent finite
difference method (S-FDM) in which the non-uniform mesh depends on the time
fractional derivative order of FRE. The purpose is to establish a stable
numerical method with low computation cost for FREs by making a bridge between
the fractional derivative order and space-time discretization steps. The
proposed method is proved to be unconditional stable with (2-{\alpha})-th
convergence rate. Moreover, three examples are carried out to make a comparison
among the uniform difference method, common non-uniform method and S-FDM in
term of accuracy, convergence rate and computational costs. It has been
confirmed that the S-FDM method owns obvious advantages in computational
efficiency compared with uniform mesh method, especially for long-time range
computation (e.g. the CPU time of S-FDM is ~1/400 of uniform mesh method with
better relative error for time T=500 and fractional derivative order
alpha=0.4).Comment: 26 pages,8 figure
Second order finite difference approximations for the two-dimensional time-space Caputo-Riesz fractional diffusion equation
In this paper, we discuss the time-space Caputo-Riesz fractional diffusion
equation with variable coefficients on a finite domain. The finite difference
schemes for this equation are provided. We theoretically prove and numerically
verify that the implicit finite difference scheme is unconditionally stable
(the explicit scheme is conditionally stable with the stability condition
) and 2nd order convergent in space direction, and
-th order convergent in time direction, where .Comment: 27 page
Finite difference/local discontinuous Galerkin method for solving the fractional diffusion-wave equation
In this paper a finite difference/local discontinuous Galerkin method for the
fractional diffusion-wave equation is presented and analyzed. We first propose
a new finite difference method to approximate the time fractional derivatives,
and give a semidiscrete scheme in time with the truncation error , where is the time step size. Further we develop a fully
discrete scheme for the fractional diffusion-wave equation, and prove that the
method is unconditionally stable and convergent with order , where is the degree of piecewise polynomial. Extensive numerical
examples are carried out to confirm the theoretical convergence rates.Comment: 18 pages, 2 figure
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