Local convergence analysis of a linearized Alikhanov scheme for the time fractional sine-Gordon equation

TL;DR

This paper develops and analyzes a linearized Alikhanov scheme for the time fractional sine-Gordon equation, establishing stability and convergence results under general mesh grading, with numerical validation of optimal convergence rates.

Contribution

It introduces a fully discrete linearized scheme for the fractional sine-Gordon equation and provides sharp error bounds and stability analysis on graded meshes.

Findings

Temporal convergence order is min{2, r} in H^1-seminorm.

Numerical experiments confirm optimal convergence rate at r=2.

Stability results are valid on general graded temporal meshes.

Abstract

This paper investigates the time fractional sine-Gordon equation whose solution exhibits a weak singularity of type t^{\alpha}. By means of the Alikhanov formula we derive a fully discrete, linearized scheme. Using the more general regularity assumption, we derive a sharp truncation-error bound for the fractional derivative. Furthermore, we prove a key inequality and a less restrictive stability result that is valid on general graded temporal meshes. Consequently, the temporal local convergence order is shown to be min{2, r} in H^1-seminorm, where r is the degree of grading; numerical experiments confirm that the optimal rate is already attained as soon as r = 2.