High-order numerical algorithms for Riesz derivatives via constructing new generating functions

TL;DR

This paper develops high-order numerical algorithms for Riesz derivatives using new generating functions, improving convergence and stability for fractional PDEs, and validates them through numerical experiments.

Contribution

Introduces a novel class of high-order algorithms for Riesz derivatives based on new generating functions, enhancing accuracy and stability over classical methods.

Findings

Achieved higher convergence orders with the new algorithms.

Established an unconditionally stable finite difference scheme for Riesz PDEs.

Numerical experiments confirm theoretical accuracy and effectiveness.

Abstract

A class of high-order numerical algorithms for Riesz derivatives are established through constructing new generating functions. Such new high-order formulas can be regarded as the modification of the classical (or shifted) Lubich's difference ones, which greatly improve the convergence orders and stability for time-dependent problems with Riesz derivatives. In rapid sequence, we apply the 2nd-order formula to one-dimension Riesz spatial fractional partial differential equations to establish an unconditionally stable finite difference scheme with convergent order $O(\tau^2+h^2)$, where $\tau$ and $h$ are the temporal and spatial stepsizes, respectively. Finally, some numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the derived numerical algorithms.