1. High order numerical methods based on quadratic spline collocation method and averaged L1 scheme for the variable-order time fractional mobile/immobile diffusion equation.
- Author
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Ye, Xiao, Liu, Jun, Zhang, Bingyin, Fu, Hongfei, and Liu, Yue
- Subjects
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COLLOCATION methods , *HEAT equation , *SPLINES , *EQUATIONS , *CRANK-nicolson method - 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 α (t) satisfies 0 < α ⁎ ≤ α (t) ≤ α ⁎ < 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 O (τ min { 3 − α ⁎ − α (0) , 2 } + Δ x 2 + Δ y 2) , where τ , Δ x and Δ y are the temporal and spatial step sizes, respectively. With some restrictions on α (t) , the QSC- L 1 + scheme has second order convergence in time 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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