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

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 L1-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 \(O\left( {\varDelta t}+{\varDelta t}^{2-\alpha }+{h}^2+{H}^\frac{5}{2}\right) \) of the proposed scheme are given as well. Numerical experiments verify the theoretical results.