1. Introduction
In this paper, we are interested in developing efficient numerical solutions of strongly nonlinear space-fractional diffusion equations (NSFDEs) of the following form:
where
are known functions. Function
may represent, for example, the concentration of a particle plume undergoing anomalous diffusion. The diffusion coefficient
is assumed to be positive, and the forcing function
represents the source or sink term [
1,
2]. In addition, we assume that Equation (
1) has a smooth solution
, and that there exist real positive numbers
and
L satisfying the following:
where
, and
. Moreover, the symbol
denotes the Riesz fractional derivative operator [
3,
4,
5] for
, defined as follows:
where
is the Gamma function.
In the past few decades, a significant amount of research work has been devoted to the analysis and solution of fractional partial differential equations (fPDEs) especially for modeling anomalous diffusion [
6,
7], subdiffusion and and superdiffusion processes that can be observed in many real physical systems, such as heat transfer problems and wave analysis [
8,
9]. Space fractional equations are obtained by replacing the traditional space derivative with a fractional derivative in the integer-order diffusion equation and can describe heterogeneous substances with memory and genetic properties very effectively [
10]. Closed-form analytic solutions are available only for limited classes of fPDEs. For solving more general equations, the only viable approach is to use discretization methods such as finite difference [
11,
12,
13,
14,
15], finite element [
16,
17], finite volume [
18,
19] and spectral [
20] methods.
The Riemann–Liouville fractional derivative is often used to model the dynamics of discrete systems with long-range interaction. Several studies have applied the weighted and shifted Grünwald difference (WSGD) formula to approximate the Riemann–Liouville derivative; see [
14,
21,
22] for details. By using the WSGD formula for two-sided fractional derivatives combining the compact technique, Hao et al. presented a fourth-order difference approximation for the space fractional derivatives [
23]. In addition, Çelik and Duman used a fractional centered difference operator to approximate the Riesz fractional derivative with second-order accuracy and to solve the Riesz SFDE (RSFDE) on a finite domain [
24]. Later, Yang et al. [
25] used the Diethelm method [
15] to derive a new
approximation for discretizing the Riesz fractional derivatives of order
, where
is the step size. Then, a new finite difference scheme for solving the RSFDE was obtained by discretizing the first-order time derivative using the Crank–Nicolson (CN) method. In contrast with the classical diffusion operator
, Riesz fractional derivative [
3,
5] is a special linear combination of left- and right-sided Riemann–Liouville fractional derivatives. By exploiting this property, Ding and Li [
4] established a novel class of high-order numerical algorithms for Riesz derivatives through constructing new generating functions. In 2018, Lin et al. [
26] studied the CN temporal discretization with various high-order spatial difference schemes for RSFDEs with variable diffusion coefficients and gave the unconditional stability and convergence analysis for temporally second-order and spatially
jth-order (
) difference schemes for such equations with variable coefficients. In 2019, Lin et al. [
27] studied the CN alternative direction implicit (CN-ADI) method for two-dimensional RSFDEs with nonseparable coefficients. They showed under mild assumptions the unconditional stability of the CN-ADI method in discrete
-norm and the consistency of cross perturbation terms arising from the CN-ADI method. We point the reader to [
3,
12,
13,
28] for other recent works on the numerical solution of fPDEs with the Riesz fractional derivative.
Due to their non-local nature, numerical approximations of space fractional operators generally lead to dense matrices. Conventional direct solution methods for linear systems based on variants of the Gaussian elimination algorithm may not be affordable to use because of their high computational complexity and large memory costs. The quest for fast algorithms for solving linear systems originated in the numerical discretization of fPDEs has become a topic of great interest in the recent literature on numerical methods for fPDEs. Lin et al. [
29] proposed a splitting preconditioner for fast solution of Toeplitz-like linear systems arising from one- and two-dimensional time-dependent RSFDEs with variable diffusion coefficients [
27]. Some other preconditioners are being applied also to solving the discretized (non-)linear systems from the solution of linear and non-linear problems; see, for example, [
1,
2,
30,
31] and the references therein.
In this paper, we study the efficient numerical solution of strongly NSFDEs because they are more suitable than linear models to describe some difficult physical processes, such as fractional distillation in nonlinear space, and when the diffusion coefficient is related to the concentration of a particle plume [
31]. However, to the best of our knowledge, only a few studies on fast numerical schemes for nonlinear or semi-linear problems have been published in the literature; see, for example, refs. [
32,
33,
34]. Our main contribution to the subject can be summarized as follows: first, for this class of strongly NSFDEs, we propose a semi-implicit difference scheme that can avoid solving the discretized nonlinear systems [
35], and analyze its stability and convergence properties. Then, we extend the method to the solution of the two-dimensional formulation of Equation (
1). In order to compute an efficient iterative solution of the discretized systems, a suitable preconditioning technique is developed and presented in the paper. The complexity and memory requirement of the resulting semi-implicit difference scheme are fairly lower, compared to the fully implicit difference scheme that requires to solve the discretized nonlinear systems, motivating our choice of the semi-implicit difference scheme for Equation (
1) [
36]. Moreover, unlike the traditional (semi-)implicit difference schemes, our proposed semi-implicit difference scheme can be implemented as a matrix-free method because it does not store any of the coefficient matrices involved in the whole solution process.
The rest of this article is organized as follows. In
Section 2, we propose a semi-implicit difference scheme for Equation (
1), and then we analyze its stability and convergence properties. In
Section 3, we extend our method to the solution of the two-dimensional formulation of Equation (
1). A fast matrix-free preconditioned iterative algorithm for solving the discretized linear system arising in the analysis of the two-dimensional case of Equation (
1) is developed, and it is presented in
Section 4. In
Section 5, we report on numerical experiments to illustrate the efficiency of the proposed semi-implicit difference method. Some conclusions arising from this work are drawn in
Section 6.
3. Extensions to the Two-Dimensional Problem
In this section, we extend our method to the solution of two-dimensional problems. For this purpose, let be positive integers and define a spatial partition of nodes with step sizes , in the x-axis and y-axis, respectively, and a uniform time partition of the interval with time step . We consider the sets of grid points , and let be a grid function defined on .
We consider the following two-dimensional problem:
where
. Upon evaluating Equation (
26) at the points
, we obtain the following discrete equation:
which can be written equivalently as follows:
Let
U be the grid function defined as follows:
Thus, the partial derivative of
t at the point
is as follows:
and
where
,
and
.
We denote for simplicity the following:
where
, so that the equation can be rewritten as follows:
with
.
Observe that the terms
are small. Omitting the quantities
in the previous expression, and imposing the initial-boundary value conditions,
the following semi-implicit difference scheme is obtained, where by
we denote the approximate solution of Equation (
31) at the points
:
Below, we analyze stability and convergence properties of the two-dimensional semi-implicit difference scheme given by Equation (
32). We define the following:
the error satisfying the following equations:
By a similar proof to Theorem 1, the following result is established about the numerical stability of the semi-implicit difference scheme given by Equation (
26).
Theorem 2. Suppose that is the solution of the semi-implicit difference scheme (25), and define the following: Then, when constants satisfy the following condition,the semi-implicit difference scheme (32) is stable and as follows: 5. Numerical Experiments
In this section, two numerical experiments are presented to illustrate the efficiency of the proposed preconditioned semi-implicit difference scheme, and to support our theoretical findings. All experiments are performed in MATLAB R2019b on Intel(R) Core(TM) i5-10210U CPU @ 1.60∼2.11 GHz and 8.00 GB of RAM. On occasion, one can reduce non-homogeneous Dirichlet boundary conditions to homogeneous Dirichlet boundary conditions by using the following transformation:
where
and
is an unknown function such that
.
Example 1. For the first problem, we consider Equation (1) with , , , , and the source term is computed as follows: This problem can be viewed as a modification of [1,24], and its exact solution is . Denoting by the numerical solution, we define the -norm errors and the -norm errors as follows:respectively, and the space and time convergence orders as follows: The data reported in
Table 1 and
Table 2 refer to the convergence results obtained by the fast preconditioned iterative algorithm and by a direct method (i.e., the backslash “∖” operator in MATLAB) for solving the non-symmetric discretized linear system, respectively. Because of the small size of the linear systems presented in
Table 2, the cost of direct solvers is affordable [
16].
Table 1 presents the maximum errors and the corresponding convergence orders in time of the semi-implicit difference scheme when
, for
, respectively. The numerical results show a convergence order approximately equal to 1 in time for the semi-implicit difference scheme. Meanwhile, the experiments reported in
Table 2 confirm that the semi-implicit difference scheme is convergent with second-order accuracy in space. These results are consistent with the theoretical analysis presented in
Section 2.
Example 2. For the second problem, we consider the two-dimensional nonlinear Riesz space-fractional diffusion problem, which is modified from [34] as follows:where , , andwhere The exact solution for this problem is .
The data reported in
Table 3 and
Table 4 are convergence results obtained with the preconditioned iterative algorithm.
Table 3 presents the maximum errors and the corresponding convergence orders in time of the semi-implicit difference scheme when
, for
, respectively. The numerical results exhibit a convergence order approximately equal to 1 in time for the semi-implicit scheme. Meanwhile,
Table 4 shows that the semi-implicit scheme is convergent with second-order accuracy in space. These results are consistent with the theoretical analysis presented in
Section 3.
Table 5 shows comparative experiments between a direct method and two fast iterative algorithms for the solution of the discretized linear system at each step of the implicit difference scheme for
. It presents the total number of iterations (Iter) and the solution time (CPU in seconds) when
. Obviously, we can see from the table that the two iterative algorithms are significantly faster than a direct algorithm, motivating our quest for fast solvers. As seen from
Figure 1, the proposed preconditioner
can be indeed efficient to accelerate the convergence of the preconditioned BiCGSTAB method in terms of the clustering eigenvalues, i.e., most eigenvalues of the preconditioned matrices increasingly gather around 1 when we take
.
Since it is proved that the convergence order of the proposed semi-implicit different scheme is , the Richardson extrapolation can be used to improve the temporal convergence order. For the sake of clarity, we do the following:
- (1)
We apply the numerical scheme (
24) with time step
, and we compute the solution
. According to the error analysis of the method, the following holds:
- (2)
Another run with the scheme (
24) is carried out using
, and a new solution
is computed on the fine temporal grid. According to the error analysis, we have the following:
- (3)
We compute
and define
:
This means that Richardson extrapolation indeed can improve the temporal convergence order of the proposed semi-implicit difference scheme from 1 to 2. However, it is always sensitive to the parameter selection used for solving the model problem and thus, it is numerically unstable (we omit the specific numerical results here).