An ADI Crank-Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation

TL;DR

This paper introduces a novel ADI Crank-Nicolson orthogonal spline collocation method for efficiently solving two-dimensional fractional diffusion-wave equations, demonstrating stability and optimal accuracy through theoretical analysis and numerical experiments.

Contribution

It presents a new combined spatial-temporal discretization approach using orthogonal splines and an ADI scheme for fractional PDEs, with proven stability and convergence.

Findings

Scheme is stable and optimally accurate.

Numerical experiments confirm convergence rates.

Superconvergence observed in results.

Abstract

A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit (ADI) method based on the Crank-Nicolson method combined with the $L1$-approximation of the time Caputo derivative of order $\alpha\in(1,2)$. It is proved that this scheme is stable, and of optimal accuracy in various norms. Numerical experiments demonstrate the predicted global convergence rates and also superconvergence.