Unconditionally stable and high-order numerical method for variable-order Caputo-Fabrizio fractional diffusion equations

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Abstract

In this work, we present and analyze a high-precision numerical scheme to solve a class of variable-order fractional diffusion equations involving the Caputo-Fabrizio (CF) fractional derivative. The proposed scheme employs a finite difference method for temporal discretization and a local discontinuous Galerkin (LDG) method equipped with generalized alternating numerical fluxes for spatial discretization. For any variable-order function $\alpha(t) \in(0,1)$, we demonstrate the stability and L² convergence of the proposed scheme. Moreover, we show that the method attains an accuracy order of O($\tau^{2}+h^{k+1}$), where τ, h, and k denote the temporal step size, spatial step size, and degree of the piecewise P^k polynomials, respectively. A series of numerical experiments are carried out to verify the theoretical findings and the effectiveness of the proposed algorithm.

Keywords

variable-order derivatives / finite difference method / local discontinuous Galerkin method / stability / error estimates

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