The Splitting Characteristic Finite Difference Domain Decomposition Scheme for Solving Time-Fractional MIM Nonlinear Advection–Diffusion Equations

• Published in: Journal of scientific computing 2024-08, Vol.100 (2), p.49, Article 49
• Main Authors: Zhou, Zhongguo, Zhang, Sihan, Li, Wanshan
• Format: Article
• Language: English
• Subjects: Advection; Advection-diffusion equation; Algorithms; Boundary conditions; Computation; Computational Mathematics and Numerical Analysis; Decomposition; Diffusion barriers; Domain decomposition methods; Finite difference method; Finite volume method; Fixed points (mathematics); Fluxes; Fractals; Interiors; Interpolation; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Numerical analysis; Splitting; Theoretical; Tracking
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-024-02603-4

In this paper, we develop a new splitting characteristic finite difference scheme for solving the time-fractional mobile-immobile nonlinear advection–diffusion equation by combining non-overlapping block-divided domain decomposition method, the operator splitting technique and the characteristic finite difference method. Over each sub-domain, the solutions and fluxes along x -direction in the interiors of sub-domains are computed by the implicit characteristic finite difference method while the intermediate fluxes on the interfaces of sub-domains are computed by local multi-point weighted average from the approximate solutions at characteristic tracking points which are solved by the quadratic interpolation. Secondly, the solutions and fluxes along y direction in the interiors of sub-domains are computed lastly by the implicit characteristic difference method while the time fractional derivative is approximated by L 1-format and the intermediate fluxes on the interfaces of sub-domains are computed by local multi-point weighted average from the approximate solutions at characteristic tracking points are solved by the quadratic interpolation. Applying Brouwer fixed point theorem, we prove strictly the existence and uniqueness of the proposed scheme. The conditional stability and convergence with O Δ t + Δ t 2 - α + h 2 + H 5 2 of the proposed scheme are given as well. Numerical experiments verify the theoretical results.