16 research outputs found
Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
In this work, we extend the fractional linear multistep methods in [C.
Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional
integral and derivative operators in the sense that the tempered fractional
derivative operator is interpreted in terms of the Hadamard finite-part
integral. We develop two fast methods, Fast Method I and Fast Method II, with
linear complexity to calculate the discrete convolution for the approximation
of the (tempered) fractional operator. Fast Method I is based on a local
approximation for the contour integral that represents the convolution weight.
Fast Method II is based on a globally uniform approximation of the trapezoidal
rule for the integral on the real line. Both methods are efficient, but
numerical experimentation reveals that Fast Method II outperforms Fast Method I
in terms of accuracy, efficiency, and coding simplicity. The memory requirement
and computational cost of Fast Method II are and ,
respectively, where is the number of the final time steps and is the
number of quadrature points used in the trapezoidal rule. The effectiveness of
the fast methods is verified through a series of numerical examples for
long-time integration, including a numerical study of a fractional
reaction-diffusion model
Highly efficient schemes for time fractional Allen-Cahn equation using extended SAV approach
In this paper, we propose and analyze high order efficient schemes for the
time fractional Allen-Cahn equation. The proposed schemes are based on the L1
discretization for the time fractional derivative and the extended scalar
auxiliary variable (SAV) approach developed very recently to deal with the
nonlinear terms in the equation. The main contributions of the paper consist
in: 1) constructing first and higher order unconditionally stable schemes for
different mesh types, and proving the unconditional stability of the
constructed schemes for the uniform mesh; 2) carrying out numerical experiments
to verify the efficiency of the schemes and to investigate the coarsening
dynamics governed by the time fractional Allen-Cahn equation. Particularly, the
influence of the fractional order on the coarsening behavior is carefully
examined. Our numerical evidence shows that the proposed schemes are more
robust than the existing methods, and their efficiency is less restricted to
particular forms of the nonlinear potentials
Efficient high order algorithms for fractional integrals and fractional differential equations
We propose an efficient algorithm for the approximation of fractional
integrals by using Runge--Kutta based convolution quadrature. The algorithm is
based on a novel integral representation of the convolution weights and a
special quadrature for it. The resulting method is easy to implement, allows
for high order, relies on rigorous error estimates and its performance in terms
of memory and computational cost is among the best to date. Several numerical
results illustrate the method and we describe how to apply the new algorithm to
solve fractional diffusion equations. For a class of fractional diffusion
equations we give the error analysis of the full space-time discretization
obtained by coupling the FEM method in space with Runge--Kutta based
convolution quadrature in time
Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system
A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims
Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives
In this paper, we consider a numerical method for the multi-term
Caputo-Fabrizio time-fractional diffusion equations (with orders
, ). The proposed method employs a fast
finite difference scheme to approximate multi-term fractional derivatives in
time, requiring only storage and computational complexity,
where denotes the total number of time steps. Then we use a Legendre
spectral collocation method for spatial discretization. The stability and
convergence of the scheme have been thoroughly discussed and rigorously
established. We demonstrate that the proposed scheme is unconditionally stable
and convergent with an order of , where
, , and represent the timestep size, polynomial degree, and
regularity in the spatial variable of the exact solution, respectively.
Numerical results are presented to validate the theoretical predictions
Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
Time-fractional partial differential equations are nonlocal in time and show
an innate memory effect. In this work, we propose an augmented energy
functional which includes the history of the solution. Further, we prove the
equivalence of a time-fractional gradient flow problem to an integer-order one
based on our new energy. This equivalence guarantees the dissipating character
of the augmented energy. The state function of the integer-order gradient flow
acts on an extended domain similar to the Caffarelli-Silvestre extension for
the fractional Laplacian. Additionally, we apply a numerical scheme for solving
time-fractional gradient flows, which is based on kernel compressing methods.
We illustrate the behavior of the original and augmented energy in the case of
the Ginzburg-Landau energy functional