Finite volume approximation with ADI scheme and low-rank solver for high dimensional spatial distributed-order fractional diffusion equations

• Published in: Computers & mathematics with applications (1987) Vol. 89; pp. 116 - 126
• Main Authors: Chou, Lot-Kei, Lei, Siu-Long
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.05.2021; Elsevier BV
• Subjects: Alternating direction implicit methods; Alternating direction implicit scheme; Approximation; Compressing; Convergence; Distributed-order; Finite volume approximation; Format; High dimensional; Initial conditions; Mathematical analysis; Perturbation methods; Quadratures; Tensor-Train format; Tensors; Tensor-Train format; High dimensional; Distributed-order; Finite volume approximation; Alternating direction implicit scheme
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2021.02.014

High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order 3−α, where α is the maximum of the involving fractional orders. Moreover, if the initial condition and source term possess Tensor-Train format (TT-format) with low TT-ranks, the scheme can be solved in TT-format, such that higher dimensional cases can be considered. Perturbation analysis ensures that the accumulated errors due to data recompression do not affect the overall convergence order. Numerical examples with low TT-ranks initial conditions and source terms, and with dimensions up to 20 are tested.