An implicit integration factor method for a kind of spatial fractional diffusion equations

A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete ordinary differential system by using the implicit integration factor method, which is a class of efficient semi-implicit temporal scheme. Numerical results show that the proposed scheme is accurate even for the discontinuous coefficients.


Introduction
In this work, a special nonlinear case of the spatial fractional diffusion equation (SFDE) proposed in [1] is studied: (1 ) (1 ) The differential equations with nonlocal derivatives known as fractional differential equations (FDEs) have been attracted many researchers' interest in recent decades. Due to the nonlocal feature of fractional derivatives, FDEs are more suitable than the traditional differential equations in the description of the anomalous diffusion phenomena. Nowadays, the applications of FDEs have been recognized in numerous fields such as the physics [3], American options pricing [4], entropy [5] and image processing [6]. It is noticeable that finding the closed-form analytical solutions of most FDEs poses a challenge for researchers. For this reason, abundant numerical methods, e.g., finite difference method [7][8][9][10], finite element method [11,12], and meshless method [13], have been proposed to solve the FDEs. Moreover, numerous fast solvers are designed based on the structures of the resulting numerical schemes, readers are suggested to refer to [7,[14][15][16][17][18][19][20] and the references therein.
All the above mentioned studies are based on the fully discrete schemes. In this work, a semi-implicit scheme is developed to approximate (1). To our best knowledge, no article considers such the semi-implicit scheme to solve SFDE (1) by employing the implicit integration factor (IIF) method [21]. Traditional integrating factor [22] or exponential time differencing methods [23][24][25] are not efficient for the systems with severely stiff reactions, because they still treat the reaction terms explicitly. To address this problem, Nie et al. [21] proposed a new class of semi-implicit schemes for stiff systems. They showed that the stability region of their methods is much larger than existing methods, who treat the reactions explicitly. Furthermore, numerical results in [21] demonstrate that their proposed schemes are accurate, robust and efficient. Later, Nie et al. [26] proposed a compact IIF method to solve the high-dimensional stiff reaction-diffusion equations. Such the method preserves the stability property of the IIF method and saves the storage requirement and CPU times. Other studies about IIF method can be found in [27][28][29][30][31].
The rest of this paper is organized as follows. In Section 2, a second-order implicit integration factor (IIF2) scheme is derived and its linear stability is also studied. Several numerical examples are provided in Section 3. Some conclusions are drawn in Section 4.

A second-order implicit integration factor scheme
In this section, the IIF2 scheme is derived and its stability is studied.

and the relationships between Caputo and
Riemann-Liouville fractional derivatives [2]: Then the semi-discrete ordinary differential system of Equation (1) is Then let With the help of these symbols, the matrix form of Equation (2) can be written as: For the discretization of the time direction, we use the IIF method proposed in [21] instead of the According to the work [21], the IIF2 scheme of (3) is where ( )

Linear stability analysis of IIF2
Similar to Section 3 in [21], the linear stability of the IIF2 scheme is studied in this subsection. Applying Equation (4) to the following scalar linear equation: In this work, the fixed-point iteration method (the maximum number of iterations and tolerance are respectively 200, 1e-12) is used to solve Equation (4). All experiments were performed on a Windows 7 (32 bit) desktop-Intel(R) Core(TM) i3-2130 CPU 3.40GHz, 4GB of RAM using Spyder 3.2.8.   Table 2 lists the maximum norm errors and illustrates that the spatial convergence order is of 2 h   . In a word, Tables 1-2 confirm that the rate of the truncation error of the IIF2 scheme (4) is   It can be seen from Tables 3-4 that the order of accuracy of the IIF2 scheme (4) is also   2 2 h      for the discontinuous coefficients. As a conclusion, our scheme is accurate for the discontinuous coefficients.

Concluding remarks
In this work, the IIF2 scheme (4) is derived by employing the 1 L formula and the IIF method, which treats the diffusion term exactly and the nonlinear reaction term implicitly, to approximate a kind of nonlinear SFDE (1). Then, the linear stability of (4) is analysed in Section 2.2. Numerical results show that the order of accuracy of our scheme is of  