A note on efficient preconditioner of implicit Runge–Kutta methods with application to fractional diffusion equations

Abstract

This paper is concerned with the construction of efficient preconditioning method for systems arising from implicit Runge–Kutta discretizations of time-dependent PDEs. A polynomial preconditioner is proposed based on the Kronecker product splitting iteration method introduced by Chen (BIT 54:607-721, 2014). Some useful properties on the spectrum of the preconditioned matrix are established. Numerical experiments concerning discrete solution of fractional diffusion equations are presented to show the effectiveness of the proposed polynomial preconditioner.

Introduction

Consider the ordinary differential equations (ODEs)

$$\{\begin{array}{c}{\displaystyle \frac{dv\left(t\right)}{dt}}=-\tilde{L}v\left(t\right)+f\left(t\right),t\in [t_0,T],\hfill \\ v\left(t_0\right)=v_0,\hfill \end{array}$$

where $v\left(t\right),f\left(t\right):[t_0,T]\to {\mathbb{R}}^N,$ $v_0\in {\mathbb{R}}^N,$ and $\tilde{L}\in {\mathbb{R}}^{N\times N}$. We are interested in the iterative solution of linear systems

$$Qx=b,Q=I_s\otimes I_N+A\otimes L$$

that arise in the numerical integration of ODEs (1.1) based on implicit Runge–Kutta (IRK) methods (see, for instance, [17]). Here $A\in {\mathbb{R}}^{s\times s}$ is the coefficient matrix of the IRK method, I_s is the s × s identity matrix, $L=\tau \tilde{L},$ τ is the integration stepsize and ⊗ denotes the Kronecker product. IRK time integration schemes exhibit attractive properties such as high order of accuracy and favorable stability properties (A-stability) and they are widely used for stiff problems. However, efficient solution strategies are not easy to design. A typical s-stage IRK method will lead to coupled s × s block systems (1.2), which may be large-scale and ill-conditioned. For the numerical solution of the systems (1.2) arising from integer-order partial differential equations (PDEs), some iterative methods and preconditioning techniques have been proposed, see, for example, [1], [2], [4], [6], [7], [8], [9], [12], [19], [24], [28], [29], [31], [32].

The aim of this paper is the construction of efficient preconditioner for systems (1.2) arising from IRK discretization of the fractional diffusion equations (FDEs). Since IRK methods lead to a coupled s × s block system which is in a form of the sum of two Kronecker products, the idea is to use one Kronecker product to approximate the summation of two Kronecker products. This Kronecker product preconditioner can be derived from a Kronecker product splitting (KPS) iteration method [6]. A KPS-based polynomial preconditioner is then proposed based on that KPS iteration method and we analyze the spectrum of the corresponding preconditioned matrix. The building block of the KPS-based preconditioner has the same structure as the system matrix derived from implicit Euler discretization of the FDEs. Therefore, we can recover the (already available) high performances of the implicit Euler discretization solvers for the IRK methods.

The remaining part of this paper is organized as follows. In Section 2 we review the KPS iteration method briefly. In Section 3 we present a KPS-based polynomial preconditioner based on the KPS iteration method and analyze the spectrum of the preconditioned matrix. In Section 4, numerical experiments concerning the numerical solutions of FDEs are presented to demonstrate the performance of the proposed preconditioner. Finally, some conclusions are given in Section 5.