Abstract
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.
Highlights
Fractional Differential Equations (FDEs) have nowadays become very popular for modeling different physical processes, such as anomalous diffusion [26] or viscoelasticity [1,25]
We provide a complete error analysis of the discretization in space and time of a class of fractional diffusion equations
For en(λ) a function which depends on the ODE method underlying the convolution quadrature (CQ) formula and an integration contour which can be chosen as a Hankel contour beginning and ending in the left half of the complex plane
Summary
Fractional Differential Equations (FDEs) have nowadays become very popular for modeling different physical processes, such as anomalous diffusion [26] or viscoelasticity [1,25]. In the present paper we develop a fast and memory efficient method to. As en(z) = r (z)nq(z) and r (z) = ez + O(z p+1), where p is the order of the underlying RK method, this is intimately related to the construction of an efficient quadrature for the integral representation of the convolution kernel t α−1 =. Our main contribution here is the development of an efficient quadrature to approximate (3) and its use in a fast and memory efficient scheme for computing the discrete convolution (2). The fast and oblivious quadratures of [19] and [23] have the same asymptotic complexity as our algorithm, but have a more complicated memory management structure and require the optimization of the shape of the integration contour. 3, we recall Convolution Quadrature based on Runge–Kutta methods and derive the special representation of the associated weights already stated in (3). We develop an efficient quadrature for (4) accurate for t ∈ [n0h, T ]
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