A fast and high-order localized meshless method for fourth-order time-fractional diffusion equations

Abstract

This paper develops a localized meshless collocation technique for solving the two-dimensional (2D) fourth-order diffusion equation with the Caputo time-fractional derivative of order α∈(0,1). An auxiliary variable is introduced to transform the original fourth-order problem into an equivalent second-order system, preserving the weak singularity of the exact solution at the initial time. Two time semi-discrete schemes are constructed: the first utilizes the standard Alikhanov formula on general time meshes, including both uniform and nonuniform meshes, to approximate the Caputo fractional derivative; the second employs the fast Alikhanov formula, derived from the sum-of-exponentials (SOEs) technique, to reduce computational costs and storage requirements. The local radial basis function partition of unity (LRBF-PU) collocation method is then used for spatial discretization, resulting in two distinct fully discrete schemes. The unconditional stability and convergence of both time semi-discrete schemes are proved by L2 and H1 energy methods. Convergence analysis reveals that the algorithms achieve optimal second-order accuracy in time by selecting an appropriate time mesh parameter γ, which influences the temporal convergence rate. Numerical results on regular and irregular spatial domains with uniform and scattered nodes demonstrate the accuracy, efficiency, and robustness of the proposed algorithms, confirming the theoretical analysis.