Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations ()
1. Introduction
In the last two decades, many fractional differential models have been studied, and these models [1] - [8] have been widely applied in many fields of science and technology, and a lot of research results have been obtained. Among them, Meerschaert and Tadjeran [4] proposed the shifted Grünwald difference operator, which combined with the Crank-Nicolson method to derive the numerical scheme of the advection-dispersion equation. The scheme is unconditionally stable and convergent with order
. Later, Meerschaert and Tadjeran [5] utilized the shifted Grünwald difference operators to approximate the left and right Riemann-Liouville fractional derivatives, in time direction is discretized by the explicit Euler method, numerical scheme is unconditionally stable and convergent with order
. Tadjeran et al. [9] adopted Crank-Nicolson method to discrete time partial derivative, and one-order shifted Grünwald difference operator to approximate space fractional derivative, derived numerical scheme with convergence order
for solving the one-sided space fractional diffusion equation with variable coefficients. By weighting the shifted Grünwald difference operator, Tian et al. [10] combined Crank-Nicolson time discretization to construct a class of second-order numerical scheme for solving the two-sided space fractional diffusion equations. The schemes are proven to achieve convergence accuracy
. Zhou et al. [11] used the compact technique on the basis of weighted and shifted Grünwald difference operator and combined Crank-Nicolson time discretization to derive quasi-compact scheme for solving the two-sided space fractional diffusion equations, which is proved to be unconditionally stable and convergence with order
. Hao et al. [12] further derived numerical scheme with convergence accuracy
by using the compact technique. Haghi et al. [7] proposed a high-order compact numerical scheme for solving the two-dimensional nonlinear time-fractional fourth-order reaction-diffusion equation, the unique solvability of the numerical method is proved in detail. In addition, Li and Deng [13] proposed tempered weighted and shifted Grünwald difference operators for the Riemann-Liouville tempered fractional derivatives, and then a class of second-order numerical schemes is proposed for solving two-sided space tempered fractional diffusion equation. Numerical schemes show convergence order is
. More work on tempered fractional models reference [14] - [23] .
In this paper, the following nonlinear fractional advection-diffusion equations are considered:
(1)
where
,
is the mean advective velocity, non-negative constants l and r denote the diffusion coefficients, which satisfy that
,
if
, and
if
, the nonlinear source term
satisfies Lipschitz condition:
(2)
and
represent the left and right Riemann-Liouville fractional derivatives respectively, which are defined as [24] :
The sections of this article are set as follows. The quasi-compact difference approximations of fractional derivatives are introduced in Section 2. The derivation process of numerical scheme for solving problem (1) is given in Section 3. Section 4 gives a detailed proof of the stability and convergence of the numerical scheme. In Section 5, numerical experiments are given to show that the numerical scheme constructed is effective. Section 6 gives a brief summary of the work.
2. Quasi-Compact Difference Approximations for the Fractional Derivatives
is a fractional Sobolev space
, where
is the Fourier transform of
.
Lemma 2.1. [9] Let
,
, the shift number p is an integer. The shifted Grünwald difference operators are defined as:
(3)
(4)
then
(5)
(6)
where
(
) denotes the normalized Grünwald weights,
are the power series expansion coefficients of the function
.
Lemma 2.2. [25] Let
,
, if two difference operators are defined as:
(7)
(8)
then
(9)
(10)
where
,
,
,
.
Noticed,
,
[24] , specially,
(11)
(12)
3. Numerical Scheme
The time interval
and the space interval
are divided into equidistant grids, and the time stepsize is denoted as
and the space stepsize is denoted as
respectively,
,
,
,
.
Denoting
,
,
,
,
.
The function
in problem (1) belongs to
after zero extension.
Applying
to both sides of the equation in problem (1), andthen discretizing time partial derivative at point
by Crank-Nicolson method, and from Lemma 2.2, we can see:
(13)
(14)
where
is the local truncation error.
Denoting
, the corresponding numerical scheme is obtained by eliminating the local truncation error in (14):
(15)
The corresponding matrix form is:
(16)
where
,
,
,
,
,
(17)
4. Theoretical Analysis of the Numerical Scheme
Before giving a detailed theoretical analysis, let’s first give some lemmas that will be used.
Lemma 4.1. [26] Real matrix A of order n is negative definite if and only if
is negative definite.
Lemma 4.2.
, for all given nonzero real column vectors
, C satisfies that:
Proof. By Gerschgorin disk theorem [27] , it is easy to check
, then
.¨
Lemma 4.3. Matrix P is given in (16) is negative definite.
Proof. Let
, from [17] ,
is negative definite, by Lemma 4.1, it is known that the matrix P is negative definite.¨
Define
is a grid function defined on
. For
, the corresponding discrete
-norm is defined as
.
Theorem 4.1. The numerical scheme (15) is stable.
Proof. Let
and
represent the numerical solutions obtained by scheme (15) under different initial conditions respectively. Denoting
,
, then
(18)
where
.
Let’s multiply both sides of (18) by
, we get:
(19)
because
(20)
arrange the formula to obtain:
(21)
The following inequalities are established:
Denoting
, from Lemma 4.3 and (21), then
(22)
Applying
, then for all given
, when
, we obtain:
(23)
By the discrete Gronwall inequalities,
(24)
Therefore,
(25)
where
.¨
Theorem 4.2. The numerical scheme (15) is convergent.
where
,
,
is existed constant.
Proof. Subtracting (15) from (14), we know that:
(26)
The corresponding matrix form is:
(27)
where
.
Let’s multiply both sides of (27) by
,
(28)
Denoting
, similar to the proof of Theorem 4.1, then
(29)
Applying
, and noticing that
, then for all given
, when
, we obtain:
(30)
By the discrete Gronwall inequalities,
(31)
(32)
¨
5. Numerical Experiments
In this section, we give some numerical experiments to prove the accuracy and validity of the numerical scheme.
is the order of measurement.
Example 5.1. Consider the following nonlinear fractional advection-diffusion equations.
where
, and the nonlinear source term is:
The analytical solution is
.
Choose different
, the mean advective velocity
, diffusion coefficients l and r, the proposed method is used to solve Example 5.1. Let
, the error results and measurement order results obtained by the numerical method are displayed in Table 1 and Table 2. From Table 1 and Table 2, we can see that the numerical scheme reaches the third-order precision in the spatial direction, and the results are in complete agreement with the conclusion of theoretical analysis. In Table 3 and Table 4, we choose different time stepsizes and fixed
, and obtain the errors and the time measurement order results. From Table 3 and Table 4, we verify the numerical scheme is second-order in time.
6. Conclusion
The novelty of this paper is that the higher-order numerical scheme of the
Table 1. Errors and the corresponding space measurement order results at
,
,
,
.
Table 2. Errors and the corresponding space measurement order results at
,
,
,
.
Table 3. Errors and the corresponding time measurement order results at
,
,
,
,
.
Table 4. Errors and the corresponding time measurement order results at
,
,
,
,
.
nonlinear fractional advection-diffusion equations is studied, and the spatial convergence accuracy reaches the third order. Firstly, Crank-Nicolson quasi-compact scheme is constructed. Secondly, the numerical scheme is analyzed theoretically by the energy method in the sense of L2-norm. Finally, the effectiveness of the numerical scheme is verified in numerical experiments.
Acknowledgments
The authors would like to express thanks to the editors and anonymous reviewers for their valuable comments and suggestions. This research is supported by the Xinjiang University of Political Science and Law President Fund (XZSK2022039).