9,772 research outputs found
Finite difference schemes for multi-term time-fractional mixed diffusion-wave equations
The multi-term time-fractional mixed diffusion-wave equations (TFMDWEs) are
considered and the numerical method with its error analysis is presented in
this paper. First, a approximation is proved with first order accuracy to
the Caputo fractional derivative of order Then the
approximation is applied to solve a one-dimensional TFMDWE and an implicit,
compact difference scheme is constructed. Next, a rigorous error analysis of
the proposed scheme is carried out by employing the energy method, and it is
proved to be convergent with first order accuracy in time and fourth order in
space, respectively. In addition, some results for the distributed order and
two-dimensional extensions are also reported in this work. Subsequently, a
practical fast solver with linearithmic complexity is provided with partial
diagonalization technique. Finally, several numerical examples are given to
demonstrate the accuracy and efficiency of proposed schemes.Comment: approximation compact difference scheme distributed order fast
solver convergenc
A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional Allen-Cahn equations
In this paper, by using Strang's second-order splitting method, the numerical
procedure for the three-dimensional (3D) space fractional Allen-Cahn equation
can be divided into three steps. The first and third steps involve an ordinary
differential equation, which can be solved analytically. The intermediate step
involves a 3D linear fractional diffusion equation, which is solved by the
Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique
can convert the multidimensional problem into a series of one-dimensional
problems, which greatly reduces the computational cost. A fourth-order
difference scheme is adopted for discretization of the space fractional
derivatives. Finally, Richardson extrapolation is exploited to increase the
temporal accuracy. The proposed method is shown to be unconditionally stable by
Fourier analysis. Another contribution of this paper is to show that the
numerical solutions satisfy the discrete maximum principle under reasonable
time step constraint. For fabricated smooth solutions, numerical results show
that the proposed method is unconditionally stable and fourth-order accurate in
both time and space variables. In addition, the discrete maximum principle is
also numerically verified.Comment: 24 pages, 7 figures, 10 table
An efficient second-order convergent scheme for one-side space fractional diffusion equations with variable coefficients
In this paper, a second order finite difference scheme is investigated for
time-dependent one-side space fractional diffusion equations with variable
coefficients. The existing schemes for the equation with variable coefficients
have temporal convergence rate no better than second order and spatial
convergence rate no better than first order, theoretically. In the presented
scheme, the Crank-Nicolson temporal discretization and a second-order
weighted-and-shifted Gr\"unwald-Letnikov spatial discretization are employed.
Theoretically, the unconditional stability and the second-order convergence in
time and space of the proposed scheme are established under some conditions on
the diffusion coefficients. Moreover, a Toeplitz preconditioner is proposed for
linear systems arising from the proposed scheme. The condition number of the
preconditioned matrix is proven to be bounded by a constant independent of the
discretization step-sizes so that the Krylov subspace solver for the
preconditioned linear systems converges linearly. Numerical results are
reported to show the convergence rate and the efficiency of the proposed
scheme
Efficient computation of the Grunwald-Letnikov fractional diffusion derivative using adaptive time step memory
Computing numerical solutions to fractional differential equations can be
computationally intensive due to the effect of non-local derivatives in which
all previous time points contribute to the current iteration. In general,
numerical approaches that depend on truncating part of the system history while
efficient, can suffer from high degrees of error and inaccuracy. Here we
present an adaptive time step memory method for smooth functions applied to the
Grunwald-Letnikov fractional diffusion derivative. This method is
computationally efficient and results in smaller errors during numerical
simulations. Sampled points along the system history at progressively longer
intervals are assumed to reflect the values of neighboring time points. By
including progressively fewer points backward in time, a temporally weighted
history is computed that includes contributions from the entire past of the
system, maintaining accuracy, but with fewer points actually calculated,
greatly improving computational efficiency.Comment: 25 pages; in press in The Journal of Computational Physic
A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term
An implicit finite difference scheme based on the - formula
is presented for a class of one-dimensional time fractional reaction-diffusion
equations with variable coefficients and time drift term. The unconditional
stability and convergence of this scheme are proved rigorously by the discrete
energy method, and the optimal convergence order in the -norm is
with time step and mesh size . Then, the
same measure is exploited to solve the two-dimensional case of this problem and
a rigorous theoretical analysis of the stability and convergence is carried
out. Several numerical simulations are provided to show the efficiency and
accuracy of our proposed schemes and in the last numerical experiment of this
work, three preconditioned iterative methods are employed for solving the
linear system of the two-dimensional case.Comment: 27 pages, 5 figures, 5 table
An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains
In this article, an advanced differential quadrature (DQ) approach is
proposed for the high-dimensional multi-term time-space-fractional partial
differential equations (TSFPDEs) on convex domains. Firstly, a family of
high-order difference schemes is introduced to discretize the time-fractional
derivative and a semi-discrete scheme for the considered problems is presented.
We strictly prove its unconditional stability and error estimate. Further, we
derive a class of DQ formulas to evaluate the fractional derivatives, which
employs radial basis functions (RBFs) as test functions. Using these DQ
formulas in spatial discretization, a fully discrete DQ scheme is then
proposed. Our approach provides a flexible and high accurate alternative to
solve the high-dimensional multi-term TSFPDEs on convex domains and its actual
performance is illustrated by contrast to the other methods available in the
open literature. The numerical results confirm the theoretical analysis and the
capability of our proposed method finally.Comment: 22 pages, 26 figure
Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations
The paper investigates a non-intrusive parallel time integration with
multigrid for space-fractional diffusion equations in two spatial dimensions.
We firstly obtain a fully discrete scheme via using the linear finite element
method to discretize spatial and temporal derivatives to propagate solutions.
Next, we present a non-intrusive time-parallelization and its two-level
convergence analysis, where we algorithmically and theoretically generalize the
MGRIT to time-dependent fine time-grid propagators. Finally, numerical
illustrations show that the obtained numerical scheme possesses the saturation
error order, theoretical results of the two-level variant deliver good
predictions, and significant speedups can be achieved when compared to parareal
and the sequential time-stepping approach.Comment: 20 pages, 4 figures, 8 table
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 methods for nonlocal and fractional models
Partial differential equations (PDEs) are used, with huge success, to model
phenomena arising across all scientific and engineering disciplines. However,
across an equally wide swath, there exist situations in which PDE models fail
to adequately model observed phenomena or are not the best available model for
that purpose. On the other hand, in many situations, nonlocal models that
account for interaction occurring at a distance have been shown to more
faithfully and effectively model observed phenomena that involve possible
singularities and other anomalies. In this article, we consider a generic
nonlocal model, beginning with a short review of its definition, the properties
of its solution, its mathematical analysis, and specific concrete examples. We
then provide extensive discussions about numerical methods, including finite
element, finite difference, and spectral methods, for determining approximate
solutions of the nonlocal models considered. In that discussion, we pay
particular attention to a special class of nonlocal models that are the most
widely studied in the literature, namely those involving fractional
derivatives. The article ends with brief considerations of several modeling and
algorithmic extensions which serve to show the wide applicability of nonlocal
modeling.Comment: Revised/Improved version. 126 pages, 18 figures, review pape
Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation
The functional distributions of particle trajectories have wide applications,
including the occupation time in half-space, the first passage time, and the
maximal displacement, etc. The models discussed in this paper are for
characterizing the distribution of the functionals of the paths of anomalous
diffusion described by time-space fractional diffusion equation. This paper
focuses on providing effective computation methods for the models. Two kinds of
time stepping schemes are proposed for the fractional substantial derivative.
The multiresolution Galerkin method with wavelet B-spline is used for space
approximation. Compared with the finite element or spectral polynomial bases,
the wavelet B-spline bases have the advantage of keeping the Toeplitz structure
of the stiffness matrix, and being easy to generate the matrix elements and to
perform preconditioning. The unconditional stability and convergence of the
provided schemes are theoretically proved and numerically verified. Finally, we
also discuss the efficient implementations and some extensions of the schemes,
such as the wavelet preconditioning and the non-uniform time discretization.Comment: 31 pages, 1 figur
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