Roundoff error problem in L2-type methods for time-fractional problems

TL;DR

This paper introduces a new framework to prevent catastrophic cancellations in L2-type methods for time-fractional problems, improving accuracy and efficiency in long-time simulations on nonuniform meshes.

Contribution

A novel concept of $oldsymbol{ ext{ extdelta}}$-cancellation and threshold conditions are proposed to avoid cancellations, along with a Taylor-expansion technique for enhanced stability.

Findings

The proposed method matches Gauss--Kronrod quadrature accuracy.

It significantly reduces computational cost compared to existing methods.

Enables long-time simulations with hundreds of thousands of steps.

Abstract

Roundoff error problems have occurred frequently in interpolation methods of time-fractional equations, which can lead to undesirable results such as the failure of optimal convergence. These problems are essentially caused by catastrophic cancellations. Currently, a feasible way to avoid these cancellations is using the Gauss--Kronrod quadrature to approximate the integral formulas of coefficients rather than computing the explicit formulas directly for example in the L2-type methods. This nevertheless increases computational cost and arises additional integration errors. In this work, a new framework to handle catastrophic cancellations is proposed, in particular, in the computation of the coefficients for standard and fast L2-type methods on general nonuniform meshes. We propose a concept of $\delta$-cancellation and then some threshold conditions ensuring that $\delta$-cancellations will not happen. If the threshold conditions are not satisfied, a Taylor-expansion technique is proposed to avoid $\delta$-cancellation. Numerical experiments show that our proposed method performs as accurate as the Gauss--Kronrod quadrature method and meanwhile much more efficient. This enables us to complete long time simulations with hundreds of thousands of time steps in short time.