Local convergence analysis of a linearized Alikhanov scheme for the time fractional sine-Gordon equation
Chang Hou, Hu Chen
TL;DR
This work analyzes the time-fractional sine-Gordon equation with Caputo derivative $D_t^{\alpha}$, $\alpha\in(1,2)$, addressing the initial-time singularity by reformulating via an auxiliary variable and applying the Alikhanov scheme on quasi-graded meshes. A linearized fully discrete method is developed using a symmetric coupled system with $\beta=\alpha/2$, discretized in time by the Alikhanov formula and in space by finite differences; a sharp, $\alpha$-robust truncation bound for the fractional derivative is proved. The authors establish a stability framework on general graded meshes and derive a convergence bound showing a local temporal order of $\min\{2,r\}$ in the $H^1$-seminorm, with the optimal rate attained at $r=2$. Numerical experiments verify the predicted rates and demonstrate robustness with respect to the fractional order. The results provide a practical, high-accuracy approach for nonlinear time-fractional diffusion-wave problems of sine-Gordon type on nonuniform temporal grids.
Abstract
This paper investigates the time fractional sine-Gordon equation whose solution exhibits a weak singularity of type t^α. 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.