One-Parameter Finite Difference Methods and Their Accelerated Schemes for Space-Fractional Sine-Gordon Equations with Distributed Delay (Journal of Computational Mathematics, 2024)
This paper deals with numerical methods for solving one-dimensional (1D) and two-dimensional (2D) initial-boundary value problems (IBVPs) of space-fractional sine-Gordonequations (SGEs) with distributed delay. For 1D problems, we construct a kind of one-parameter finite difference (OPFD) method. It is shown that, under a suitable condition,the proposed method is convergent with second order accuracy both in time and space.In implementation, the preconditioned conjugate gradient (PCG) method with the Strangcirculant preconditioner is carried out to improve the computational efficiency of the OPFDmethod. For 2D problems, we develop another kind of OPFD method. For such a method,two classes of accelerated schemes are suggested, one is alternative direction implicit (ADI)scheme and the other is ADI-PCG scheme. In particular, we prove that ADI scheme canarrive at second-order accuracy in time and space. With some numerical experiments, thecomputational effectiveness and accuracy of the methods are further verified. Moreover, forthe suggested methods, a numerical comparison in computational efficiency is presented.
Keywords: Fractional sine-Gordon equation with distributed delay; One-parameter finite difference methods; Convergence analysis; ADI scheme; PCG method.