178 research outputs found
Fast and oblivious convolution quadrature
We give an algorithm to compute steps of a convolution quadrature
approximation to a continuous temporal convolution using only
multiplications and active memory. The method does not require
evaluations of the convolution kernel, but instead evaluations of
its Laplace transform, which is assumed sectorial.
The algorithm can be used for the stable numerical solution with
quasi-optimal complexity of linear and nonlinear integral and
integro-differential equations of convolution type. In a numerical example we
apply it to solve a subdiffusion equation with transparent boundary conditions
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
Generalized convolution quadrature for the fractional integral and fractional diffusion equations
We consider the application of the generalized Convolution Quadrature (gCQ)
of the first order to approximate fractional integrals and associated
fractional diffusion equations. The gCQ is a generalization of Lubich's
Convolution Quadrature (CQ) which allows for variable steps. In this paper we
analyze the application of the gCQ to fractional integrals, with a focus in the
low regularity case. It is well known that in this situation the original CQ
presents an order reduction close to the singularity. Moreover, the available
theory for the gCQ does not cover this situation. Here we deduce error bounds
for a general time mesh. We show first order of convergence under much weaker
regularity requirements than previous results in the literature. We also prove
that uniform first order convergence is achievable for a graded time mesh,
which is appropriately refined close to the singularity, according to the order
of the fractional integral and the regularity of the data. Then we study how to
obtain full order of convergence for the application to fractional diffusion
equations. For the implementation of this method, we use fast and oblivious
quadrature and present several numerical experiments to illustrate our
theoretical results.Comment: 22 pages, 18 figure
A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations
A scheme for approximating the kernel of the fractional -integral
by a linear combination of exponentials is proposed and studied. The scheme is
based on the application of a composite Gauss-Jacobi quadrature rule to an
integral representation of . This results in an approximation of in an
interval , with , which converges rapidly in the number
of quadrature nodes associated with each interval of the composite rule.
Using error analysis for Gauss-Jacobi quadratures for analytic functions, an
estimate of the relative pointwise error is obtained. The estimate shows that
the number of terms required for the approximation to satisfy a prescribed
error tolerance is bounded for all , and that is bounded
for , , and
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