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

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.

32 pages, 2figures

References in corpus (1)

• A new difference scheme for the time fractional diffusion equation

Cited by in corpus (1)

• A new second-order midpoint approximation formula for Riemann-Liouville derivative: algorithm and its application