A fast method for variable-order space-fractional diffusion equations
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A fast method for variable-order space-fractional diffusion equations Jinhong Jia1 · Xiangcheng Zheng2 · Hongfei Fu3 · Pingfei Dai4 · Hong Wang2 Received: 5 July 2019 / Accepted: 3 January 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We develop a fast divide-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical scheme does not have a Toeplitz structure. In this paper, we derive a fast approximation of the coefficient matrix by the means of a finite sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O(N log2 N) memory and O(N log3 N) computational complexity with N being the numbers of unknowns. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method. Keywords Variable-order space-fractional diffusion equation · Collocation method · Divide-and-conquer algorithm · Toeplitz matrix Mathematics Subject Classification (2010) 65F05 · 65M70 · 65R20
1 Introduction Field tests showed that space-fractional diffusion equations (sFDEs) provide more accurate descriptions of challenging phenomena of superdiffusive transport and longrange interaction, which occur in solute transport in heterogeneous porous media and other applications, than integer-order diffusion equations do [4, 8, 25, 29]. In fact, integer-order diffusion equations were derived if the underlying independent and identically distributed particle movements have (i) a mean free path and (ii) a mean waiting time. In this case, the central limit theorem concludes that the (normalized) partial sum of the independent and identically distributed particle movements Hong Wang
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Numerical Algorithms
converges to Brownian motions. The probability density distribution of finding a particle somewhere in space is Gaussian, which satisfies the classical Fickian diffusion equation [24, 25]. Note that assumptions (i) and (ii) hold for diffusive transport of solute in homogeneous porous media, where solute plumes were observed to decay exponentially [2, 3] and so can be described accurately by integer-order diffusion equations. However, field tests showed that solute transport in heterogeneous aquifers often exhibits highly skewed and power-law decaying behavior, while sFDEs were derived under the assumption that the solutions have such behavior [4, 24, 25]. This is why sFDEs can accurately describe the solute transport in heterogeneous media more accurately than integer-order diffusion equations do. Consequently, they have attracted extensive research activities in the last few decades [13, 18, 19, 23]. However, sFDEs present new mathematical and numerical issues that are not common in the c
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