High order numerical methods based on quadratic spline collocation method and averaged L1 scheme for the variable-order time fractional mobile/immobile diffusion equation

Abstract

In this paper, we consider the variable-order time fractional mobile/immobile diffusion (TF-MID) equation in two-dimensional spatial domain, where the fractional order $\alpha (t)$ satisfies $0<{\alpha }_{\ast }\le \alpha (t)\le {\alpha }^{\ast }<1$. We combine the quadratic spline collocation (QSC) method and the $L{1}^{+}$ formula to propose a QSC-$L{1}^{+}$ scheme. It can be proved that, the QSC-$L{1}^{+}$ scheme is unconditionally stable and convergent with $\mathcal{O}({\tau }^{\min \{3-{\alpha }^{\ast }-\alpha (0),2\}}+\Delta x^2+\Delta y^2)$, where $\tau$, $\Delta x$ and $\Delta y$ are the temporal and spatial step sizes, respectively. With some proper assumptions on $\alpha (t)$, the QSC-$L{1}^{+}$ scheme has second temporal convergence order even on the uniform mesh, without any restrictions on the solution of the equation. We further construct a novel alternating direction implicit (ADI) framework to develop an ADI-QSC-$L{1}^{+}$ scheme, which has the same unconditionally stability and convergence orders. In addition, a fast implementation for the ADI-QSC-$L{1}^{+}$ scheme based on the exponential-sum-approximation (ESA) technique is proposed. Moreover, we also introduce the optimal QSC method to improve the spatial convergence to fourth-order. Numerical experiments are attached to support the theoretical analysis, and to demonstrate the effectiveness of the proposed schemes.

37 pages