Computationally Efficient Hybrid Method for the Numerical Solution of the 2D Time Fractional Advection-Diffusion Equation

In this paper, a hybrid method based on the Laplace transform and implicit finite difference scheme is applied to obtain the numerical solution of the two-dimensional time fractional advection-diffusion equation (2D-TFADE). Some of the major limitations in computing the numerical solution for fractional differential equations (FDEs) in multi-dimensional space are the huge computational cost and storage requirement, which are O(N) cost and O(MN) storage, N and M are the total number of time levels and space grid points, respectively. The proposed method reduced the computational complexity efficiently as it requires only O(N) computational cost and O(M) storage with reasonable accuracy when numerically solving the TFADE. The method’s stability and convergence are also investigated. The Results of numerical experiments of the proposed method are obtained and found to compare well with the results of existing standard finite difference scheme. KeywordsFractional advection-diffusion equation, Laplace transform, Finite difference scheme, Stability, Convergence.


Introduction
Fractional differential equations (FDEs) have received much interest over the last two decades due to its wide applications in many scientific branches. A detailed discussion of fractional calculus and its applications can be found in (Oldham and Spanier, 1974;Samko et al., 1993;Miller and Ross, 1993;and Podlubny, 1999). Here, the TFADE is considered as a useful tool for describing the transport dynamics in complex systems which are governed by anomalous diffusion and nonexponential relaxation patterns (Zhuang et al., 2011).
In recent years, several finite difference methods have been developed to solve fractional advection-diffusion equations (FADEs). Chen et al. (2008) presented implicit and explicit finite difference schemes for solving the fractional reaction-subdiffusion equation. The schemes had been proven to be stable and convergent using a Fourier analysis. Shen et al. (2011) established numerical explicit and implicit difference schemes for the solution of the space-time Riesz-Caputo FADE. They proved that the explicit scheme is conditionally stable, whereas the implicit scheme is unconditionally stable. Later, Gao and Sun (2015) proposed a combined compact difference scheme to solve the TFADE in one dimension. The time fractional derivative was defined in the Caputo sense. The Fourier analysis technique was employed to prove the stability and convergence of the method. Balasim and Ali (2017) derived standard point and explicit group iterative finite difference schemes to compute the numerical solution of the TFADE. The group methods were proved to be stable and convergent using matrix analysis. Mardani et al. (2018) suggested a numerical method based on finite difference scheme and meshless method for solving the onedimensional TFADE. Vong et al. (2018) presented a second order finite difference schemes to solve FADE. They utilized weighted and shifted Grunwald-Letnikov formulas to discretize the Riemann-Liouville fractional derivatives. Recently, Zhang et al. (2019) suggested implicit and explicit difference schemes to solve the time-space FADE in one dimension. Both schemes have first order accuracy in time and space.
One of the major challenges in computing the numerical solution for time fractional differential equations (TFDEs) is the nonlocal property of the fractional operator. This means that the computed values of the solution on all preceding time levels need to be stored to obtain the solution values on the present time level. This needs a substantial size of memory, especially for multi-dimensional problems. On the other hand, the computational cost of numerically solving the TFDEs using standard finite difference schemes is of ( 2 ) (Gong et al., 2014;Gong et al., 2015;Jiang et al., 2017) which is computationally very expensive and time consuming. For more information about the computational complexity of FDEs, please refer to (Gong et al., 2015). On the contrary, the computational cost of numerically solving time-dependent partial differential equations (PDEs) is of ( ), and the solution outcomes need to be saved on a fixed number of time levels (Gong et al., 2014;Jiang et al., 2017). The main aim of this study is to obtain an economical computational solution for the 2D TFADE (1). The Laplace transform will be employed to approximate the time fractional derivative and reduce the TFADE (1) to its corresponding PDE. The implicit difference scheme will then be applied to solve the resulting PDE, which reduces the computational work and memory usage significantly.
The rest of this paper is arranged as follows. Basic preliminaries and definitions are given in section in section 2. In section 3, we explain the formulation of the hybrid method for the problem (1). Unconditional Stability and convergence of the numerical scheme are proven in section 4. In section 5, we briefly review an existing standard finite difference scheme for numerically solving the TFADE (1). The computational complexities of both standard and hybrid methods are also discussed in this section. Section 6 explores the performance of the hybrid method versus the standard scheme through numerical experiments. Finally, conclusions are illustrated in section 7.

Preliminaries
Here we introduce some basic definitions and preliminaries which are utilized further in this paper. Definition 2.1 (Podlubny, 1999). The Caputo fractional derivative of order of a function (•) is given by Definition 2.2 (Zill, 2012). For ≥ 0, the Laplace transform of a function (•) is given by where, ( ) is the Laplace transform of ( ).

Formulation of the Hybrid Method
The nonlocal property of the time fractional derivative in (1) necessitates the solution outcomes on all the preceding time steps to be saved to obtain the solution on the current time step. To overcome this issue, we convert the TFADE (1) to a partial differential equation (PDE) by approximating the Caputo time fractional derivative using the Laplace transform method and linearization property proposed by Ren et al. (2016) and utilized by Bishehniasar et al. (2017) for an analytic solution as follows ℒ{ ( , , ) where, ( , , ) is the Laplace transform of ( , , ).
Define , as the numerical solution at the point ( , , ). Based on the forward in time and centered in space discretizations about the point ( , , ), the implicit finite difference scheme for solving (8) is given as After some rearrangement and neglecting the higher order terms, the following implicit finite difference approximation is obtained , in which 1 ≤ ≤ − 1, 1 ≤ ≤ − 1, 0 ≤ ≤ − 1, = 1 ℎ 2 , = 2 ℎ 2 ,

Stability and Convergence
In this section, we utilize the Fourier method  to prove the stability and convergence of the numerical scheme (12). Suppose that , is the approximate solution of the exact solution , of Eq. (12). Then the error is represented as Next, Eq. (12) can be written by error form as For = 0,1, … , − 1, define the following grid function where, ℰ ( , ) can be expanded in Fourier series as From the Parseval equality and 2 norm, we have (20) Based on the analysis above, suppose that where, = 2 1 / , = 2 2 / .
Next, we utilize an analogous approach as that used to discuss the stability to prove the convergence of the implicit scheme.
Then, we define the following discrete functions for = 0,1, … , − 1 By utilizing the 2 norm and the Parseval equality, we have Based on the analysis above, we assume that the solution of Eq. (30) has the following forms , = ℎ ℎ , , = ℎ ℎ .
where. 1 and 2 are mentioned before.

Review of a Standard Scheme
In this section, an existing standard implicit finite difference scheme for solving (1) is introduced for comparison. As stated in section 1, solving TFDEs using standard finite difference methods is time-memory consuming, especially for multi-dimensional problems. A significant reason for this is the considerable computational complexity of those methods. Consequently, the computational complexity for both standard and present methods would be analyzed in this section. The computational complexity is determined by the total number of arithmetic operations to be performed accompanied by the memory requirement for each method.
As can be seen from Eq. (47), computing the numerical solution at each unknown time level necessitates the storage of the solution values at all previous time levels. This means that there are solution values to be saved in the memory. With 8 bytes required to store each value, and disregard the memory usage of the coefficients, initial condition and source term, the standard method (47) requires about 8 bytes memory space. For instance, it requires 800 GB with = 10240, = 10240 and = 1024. According to Eq. (12), conversely, one can see that the solution requires to be stored only at two-time levels, and + 1. This indicates that we have only 2 solution outcomes to be stored in the memory. Therefore, the hybrid method needs only 16 bytes of memory space. So, the hybrid method's memory requirement is of ( ) which is much smaller than the standard method of ( ), where = .

Numerical Experiments
In this section, computational experiments are provided to illustrate our considerations. Standard scheme (47) and hybrid method developed in section 3 are considered to solve problem (1). The accuracy of these methods is evaluated by maximum absolute error and average absolute error. The hybrid method is shown to be precise and computationally efficient tool for computing the numerical solution of the TFADE (1). The numerical experiments are implemented in Mathematica 11.3 software and it is run on Windows 10 operated by Intel Quad Core Processor with 8 GB of RAM.
The maximum error between the numerical solution and the exact solution of (1) is defined as follows: Example 1. We solve the following two-dimensional problem We use Laplace transform to approximate Caputo time fractional derivative. Hence, the above problem is converted to the following system: (1, , ) = (sin + sin 1), For the numerical solution of this problem, the solution domain is discretized for different time steps of 10, 15, 20 and 25 for the time discretization (0 < < 1), and for space discretization, we assume ℎ = ℎ = 1/40 in both and directions.  In the second example, we use various time steps of 12, 16, 20 and 24 for the time discretization (0 < < 1), and for space discretization, we assume ℎ = ℎ = 1/45 in both and directions.  Figure 2, it is noted that the CPU time taken for the hybrid method to be implemented is significantly less compared to the standard method. This verifies that the hybrid method has less computational complexity than the standard method.

Conclusion
In this paper, the Laplace transform is combined with a fully implicit finite difference scheme for solving the 2D-TFADE. The proposed method has the utility of low computational complexity as it requires only ( ) memory space and ( ) computational cost, where and are the total number of time levels and space grid points, respectively. A comparison between a standard implicit scheme and the proposed method has revealed that the proposed method is accurate and superior to the standard method, in terms of the computational time and storage requirement. The unconditional stability and convergence of the numerical scheme are proved using the Fourier series analysis