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The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs. [ABSTRACT FROM AUTHOR]

The time-fractional Cattaneo equation is an equation where the fractional order α ∈ (1 , 2) has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space's semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1 σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1 σ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme. [ABSTRACT FROM AUTHOR]

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α ∈ (0 , 1) . Considering the weak singularity of the solution u (x , t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t → 0 + albeit the function itself being right continuous at t = 0 , two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2- 1 σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

• This is the first attempt to construct tailored finite point method for subdiffusion equation with anisotropic and discontinuous diffusivity. • The most distinguished feature of this new scheme is that it not only can ensure high accuracy and stability for solving subdiffusion equations with high anisotropic diffusivity but also can efficiently resolve the high gradients near the side internal. • The unique solvability and unconditional stability of proposed scheme are strictly proved. In this paper, we first propose a tailored finite point method (TFPM) for solving time fractional subdiffusion problems with anisotropic and discontinuous diffusivity. This numerical scheme can perfectly capture the rapid transition of the solutions which contain sharp interface layers even with coarse meshes. Second, the accuracy and stability of the proposed scheme are strictly analyzed. Finally, some numerical examples are provided to show the accuracy and reliability of this new scheme. [ABSTRACT FROM AUTHOR]