Fast and Parallel Runge-Kutta Approximation of Fractional Evolution Equations

TL;DR

This paper introduces a fast, parallel convolution quadrature method based on Runge-Kutta schemes for efficiently solving fractional evolution equations with sectorial operators, achieving high accuracy with reduced computational complexity.

Contribution

It develops a novel convolution quadrature algorithm using $L$-stable Runge-Kutta methods for stable, efficient approximation of fractional evolution equations, enabling parallel computation.

Findings

Algorithm computes solutions after N steps with O(N) Runge-Kutta steps.

Solution accuracy can be arbitrarily controlled by parameter ε.

Numerical examples demonstrate the method's efficiency and stability.

Abstract

We consider a linear inhomogeneous fractional evolution equation which is obtained from a Cauchy problem by replacing its first-order time derivative with Caputo's fractional derivative. The operator in the fractional evolution equation is assumed to be sectorial. By using the inverse Laplace transform a solution to the fractional evolution equation is obtained which can be written as a convolution. Based on $L$-stable Runge-Kutta methods a convolution quadrature is derived which allows a stable approximation of the solution. Here, the convolution quadrature weights are represented as contour integrals. On discretising these integrals, we are able to give an algorithm which computes the solution after $N$ time steps with step size $h$ up to an arbitrary accuracy $\varepsilon$. For this purpose the algorithm only requires $\mathcal{O}(N)$ Runge-Kutta steps for a large number of scalar linear inhomogeneous ordinary differential equations and the solutions of $\mathcal{O}(\log(N)\log(\frac{1}{\varepsilon}))$ linear systems what can be done in parallel. In numerical examples we illustrate the algorithm's performance.