Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

Abstract

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 L2-1σ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1σ 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.

Adaptive-Coefficient Finite Difference Frequency Domain Method for Solving Time-Fractional Cattaneo Equation with Absorbing Boundary Condition

“…The time-fractional Cattaneo (TFC) equation can be well applied to formulate the anomalous dynamics in the physical diffusion processes, such as dynamic crossover behaviors (see [1][2][3]). In general, the established studies for TFC equation can be classified into two categories.…”

“…In [12], Chen and Nong proposed to discretize the TFC equation by applying the Galerkin finite element method for spatial derivatives and applying backward Euler and backward difference for temporal convolution quadrature. In [3], Nong et al proposed a compact difference method for the solution to the 2D TFC equation, which can achieve a temporal accuracy of the second order and a spatial accuracy of the fourth order.…”