Roundoff error problems in interpolation methods for time-fractional problems

Abstract

Roundoff errors often disrupt interpolation methods for time-fractional equations, potentially causing suboptimal convergence or even failure. These issues primarily result from catastrophic cancellations. To address this, we introduce a novel framework for computing coefficients in standard and fast interpolation methods on nonuniform meshes. We propose δ-cancellation and associated threshold conditions to prevent such cancellations. If the thresholds aren't met, a Taylor expansion technique can be applied. Numerical experiments demonstrate our method's accuracy, on par with the Gauss–Kronrod quadrature, but significantly more efficient, allowing for extensive simulations with hundreds of thousands of time steps.