with the boundary and initial conditions
where and , the diffusion coefficients , are positive constants, is a known sufficiently smooth function, satisfies the Lipschitz condition
here is Lipschitz constant, and Riesz fractional derivatives and are defined by
where () is Gamma function.
here and are of the form
is represented as the Fourier transformation ofand defined by
In recent years, two-dimensional Riesz space fractional nonlinear reaction-diffusion equation plays an essential role in describing the propagation of the electrical potential in heterogeneous cardiac tissue (19; 20; 33; 21; 22), it attracts many author’s attention in constructing numerical methods for problems of the form (1.1)-(1.3). For approximation of Riesz derivative, Meerschaert and Tadjeran (5) initially proposed the shifted Grünwald-Letnikov approximation with first-order accuracy for Riemann-Liouville fractional derivative. Based on this approximation, Tian et al. (7) estabilished a second-order weighted and shifted Grünwald-Letnikov approximation for Riemann-Liouville fractional derivative, and the approximation was applied in Riesz space fractional advection-dispersion equations (6). Hao et al. (18) constructed a class of new weighted and shifted Grünwald-Letnikov approximation with second-order accuracy, and it was applied in (8) for fractional Ginzburg-Landau equation. Ortigueira (14)
initially proposed the fractional centered difference method with second-order accuracy for Riesz fractional derivative, and this method was applied in Riesz space fractional partial differential equation(15; 3; 21; 32; 24; 25; 16). Ding and Li (9) proposed a novel second-order approximation for Riesz derivative via constructing a new generating function, and this second-order approximation was adopted in (2) for two-dimension Riesz space-fractional diffusion equation. Recently, compact difference operator has been focused on the fractional differential equations for increasing the spatial accuracy. Zhou et al. (17) constructed a third-order quasi-compact difference scheme for Riemann-Liouville fractional derivative. Hao et al. (18) and Zhao et al. (1) proposed fourth-order compact difference operators to approximate Riemann-Liouville and Riesz derivatives, respectively, these compact difference operators have a great contribution on promoting algorithm accuracy. During these years, there also has developed some approximations by finite element method (22; 28; 29), spectral method (20; 33; 34) et al.. As we noticed, for the approximation of first-order time derivative, implicit Euler method (31; 30; 19; 28), Crank-Nicolson method (32; 22; 20; 18; 15; 6; 2), implicit midpoint method (8; 26) and BDF2 method (27; 3; 4)
are usually used, Padé approximations which are based on Runge-Kutta method are also used in recent researches(25; 24). And these methods have their own advantages for time-dependent problems.
There are some researches (19; 21; 20; 22; 33; 23; 34) on problem (1.1)-(1.3). Liu et al. (19; 21) constructed two ADI finite difference schemes, where Riesz space derivatives were discretized by shifted Grünwald-Letnikov formulae and fractional centered difference operators, respectively, implicit Euler method was applied to discretize time partial derivative, two proposed methods were proven to be stable and convergent. Bueno-Orovio et al. (33) used Fourier spectral method to approximate Riesz space fractional derivative, implicit Euler method was adopted to discretize first-order time partial derivative, a semi-implicit Fourier spectral method was developed. Zeng et al. (20) and Bu et al. (22) applied Galerkin-Legendre spectral method and Galerkin finite element method to approximate Riesz fractional derivative, respectively, two Crank-Nicolson ADI methods were established. Lin et al. (34) used a bivariate polynomial based on shifted Gegenbauer polynomials method to approximate Riesz space fractional derivative, Runge-Kutta method of order 3 was applied to discretize the first-order time partial derivative, a Runge-Kutta Gegenbauer spectral method was constructed. Iyiola et al. also discussed several implicit-explicit schemes in (23). Because the computing scale of two dimensional diffusion equation problem is very big, so a more efficient algorithm is needed. So far, there are many high-order algorithms for Riesz fractional derivative, we noticed that the fourth-order fractional compact difference operator in (1) is symmetric positive definite under certain circumstance, it’s helpful for us to analyze the stability and convergence. As we know, before ADI method is applied for solving two-dimensional nonlinear reaction-diffusion equation problem. the nonlinear source term needs to have linearized approximation. Therefore, how to deal with the nonlinear source term via linearized approximations (19; 21; 35; 27) plays an important role in constructing ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. The objective of this paper is to try to use BDF method and the fourth-order fractional compact difference operator to construct a class of new high accuracy ADI scheme based on the first-order (19) and second-order (35) linearized approximations for nonlinear source term. Stability and convergence analysis are given by energy method.
The outline of this paper is organized as follows. In 2, the numerical method is constructed for problem (1.1)-(1.3). Then in 3, stability and convergence are discussed, respectively. In 4, we use the proposed method and these methods in literatures (19; 21) to solve the test problems. Numerical results show that the proposed method has high accuracy and efficiency.
2 Numerical method
Let , , , where and are spatial step sizes, denotes time step size. and are exact solution and numerical solution of the problem (1.1)-(1.3) at , respectively. We also denote , and the boundary grid mesh is .
To discretize the Riesz space fractional derivative, we would introduce the centred difference operators which are defined by (14)
where the coefficients are determined by
then we have following lemma.
(see (1).) If , , for the fixied step-sizes and , it holds that
where the Fourth-order compact operators and are defined as follows
Before approximating the first-order partial derivative, we would introduce the properties of BDF operator.
(see (3).) For any positive integer , if , then
The nonlinear source term can be treated by following process
At the point () (1.1) becomes
where and .
Adding a small error term on both side of (2.10), we have
where and . And there exists the positive constants and such that
the boundary and initial conditions are
Multipling by in (2.13), and factorizing it, we have
where , and we can also rewrite it as follows
3 Stability and convergence analysis
In order to analyze the stability and convergence of the method, we introduce some notations and lemmas.
For any , we define the following discrete inner product and corresponding norm
In addition, for any , we denote , it implies , and denote and .
It is easy to check that and are symmetric positive definite and self-adjoint (1), Following from Lemma 3.11 in (1), one can prove that there exists the fractional symmetric positive definite difference operators and such that and , here, and are also commutable.
(see (1).) For any it holds that
where , are represented as fractional symmetric positive definite difference operators such that , .
It is easy to verify that
For any it holds that
From 3.5, we have
Similarly, we can obtain
The proof is completed. ∎
(Discrete Bellman Inequality) Let are a series of nonnegative real numbers, satisfying
then it holds that