Search

Variable-order time-fractional diffusion equations (VO-tFDEs), which can be used to model solute transport in heterogeneous porous media are considered. Concerning the well-posedness and regularity theory (cf., Zheng & Wang, Anal. Appl., 2020), two finite difference ADI and compact ADI schemes are respectively proposed for the two-dimensional VO-tFDE. We show that the two schemes are unconditionally stable and convergent with second and fourth orders in space with respect to corresponding discrete norms. Besides, efficiency and practical computation of the ADI schemes are also discussed. Furthermore, the ADI and compact ADI methods are extended to model three-dimensional VO-tFDE, and unconditional stability and convergence are also proved. Finally, several numerical examples are given to validate the theoretical analysis and show efficiency of the ADI methods. [ABSTRACT FROM AUTHOR]

We apply the dual algorithm of Chambolle for the minimization of the LLT model. A convergence theorem is given for the proposed algorithm. The algorithm overcomes the numerical difficulties related to the non-differentiability of the LLT model. The dual algorithm is faster than the original gradient descent algorithm. Numerical experiments are supplied to demonstrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]