1. Introduction
In this paper, we are interested in the robust high-order numerical methods for the following time-fractional mobile/immobile transport equation [
1]:
which is subject to the boundary condition
on
and the initial condition
in
. Here,
is a bounded convex polygonal domain and the parameters
, and
are positive constants. The source term
f and the initial condition
v are both given functions. The notation
with
denotes the Riemann–Liouville derivative defined by
In recent years, Equation (
1) has received extensive attention, and it has significant applications in diverse fields, especially in the modeling of groundwater solutes [
2,
3,
4]. There are some numerical studies for Equation (
1) or its variants; see [
1,
5,
6,
7,
8,
9] and the references therein.
In our recent paper [
1], we studied the solution regularity of Equation (
1) and proposed two efficient finite element schemes by employing convolution quadrature based on the backward Euler and the second-order backward difference methods. It is proved that the proposed schemes are robust with respect to data regularity, including the nonsmooth initial data, i.e.,
. One of the distinct features of the two schemes is that they do not impose any compatibility conditions on the source term
f, so this avoids some unrealistic assumption for the solution regularity, i.e., the assumption of the sufficiently smooth solution. When dealing with the nonsmooth solution problem, we observe numerically that the convolution quadrature generated by the high-order backward difference formula (denoted by BDF) is only of first-order accuracy when no corrections are added. This motivates us to investigate how to choose some suitable corrections in order to restore the desired high-order accuracy.
Since the convolution quadrature has a nice stable discrete structure, for which it is easy to perform the analysis of the resulting numerical scheme [
10,
11], it seems that it is suitable for developing high-order numerical schemes of the fractional model with the Riemann–Liouville derivative. In order to restore the high-order accuracy, one usually needs to add some corrections to the original scheme. There are a few papers on this direction [
12,
13,
14,
15,
16,
17,
18], just to name a few. One may refer to two review papers [
19,
20] for more details. In [
14,
16,
17], the convolution quadratures generated by the backward Euler and the second-order backward difference methods were applied to obtain the robust scheme for the time-fractional Cattaneo equation, the two-term time-fractional diffusion-wave equation, and the modified anomalous subdiffusion equation, respectively. In [
12], the authors constructed and analyzed an efficient fractional Crank–Nicolson finite element scheme with two corrections at the starting time levels. They proved that the proposed scheme is of second-order accuracy in time for both smooth and nonsmooth data. Recently, Wang et al. improved the results by adding only one single correction at the starting time level and presented the optimal error estimates of their resulting scheme [
18]. In [
13], Jin et al. developed the BDF
k (with the
k denoting the order of BDF) convolution quadrature with suitable correction terms to deal with the temporal discretization of the fractional evolution equation. Later on, Shi and Chen extended the BDF
k convolution quadrature to numerically solve the fractional Feynman–Kac equation with Lévy flight [
15]. As far as we know, there is no convolution quadrature generated by high-order BDF to solve Equation (
1) with smooth and nonsmooth problem data. The goal of this paper is to fill this gap.
The contributions of this paper are listed below. First, based on the convolution quadrature generated by BDF3/BDF4, we develop the robust temporal third/fourth-order finite element scheme for problem (
1) by carefully choosing the suitable corrections at the starting two/three steps. Second, the error estimates of the resulting schemes are expressed in term of data regularity, and are proved rigorously, cf. Theorems 2–5. Third, the designed numerical tests show that the two schemes indeed yield the temporal third/fourth-order theoretical accuracy, so this further verifies numerically the correctness of the error estimates and illustrates the significant improvement in the temporal accuracy of our schemes compared to that of [
1], cf. Tables 4–6 in
Section 6.
The rest of this paper is organized as follows. In
Section 2, we propose the temporal third-order finite element scheme based on the convolution quadrature generated by BDF3. In
Section 3, the method to obtain the suitable correction terms is discussed in detail. In
Section 4, we prove that the BDF3 is of third-order accuracy, with respect to smooth and nonsmooth data. The generalization of the BDF3 is given in
Section 5, and numerical examples are provided in
Section 6 to verify the effectiveness of the two proposed numerical schemes. Finally, we present the conclusion in the last section, i.e.,
Section 7. The symbol
c in this paper denotes a positive constant that is independent of the temporal and spatial step sizes.
3. Determination of the Two Corrections
In this part, we determine the four parameters
in the two corrections (
6) and (
7) based on the comparison of the semidiscrete solution in (
5) and the fully discrete solution in (
4). Let the notation
be the Laplace transform of
, i.e.,
. Denote the kernel
as follows:
Define the generating function with a given sequence . We have the following results for the integral representations.
Lemma 1. The semidiscrete solution in (
5)
and the fully discrete solution in (
4)
(with replacing ) have the following integral representations: Here, the contours with and and .
Proof. By using the Laplace transform technology, we can easily obtain the integral representation (
9) from (
5). One may refer to [
1] for further details.
It remains to derive the fully discrete solution in (
4) based on discrete Laplace transform, i.e., generating function.
Let
. The scheme (
4) has the following equivalent form:
with the two initial corrections (
6) and (
7). Multiplying
on both sides of (
11), and summing up for
, we obtain the following:
Noting
, one has
Thus,
from which we have
with
and the notation
. Here, we have used the change of variables
in the second equality of (
14). Denote the complex region as
enclosed by
, and
, where
. Note that the integrand
in (
14) is analytic with respect to
z in
, so applying the Cauchy’s theorem, we have
This leads to the desired result (
10) since there holds
All this completes the proof of the lemma. □
From the representations of (
9) and (
10), it is intuitive to select
in (
10) such that
and
for any given
.
For the left hand sides of (
15) and (
16), the triangle inequality leads to
and
We shall need the following lemma.
Lemma 2. Let and be defined as in (8) and (3) with , respectively. Then, for any , we have Proof. Since
, we have
In view of Lemma B.1. in [
13], we derive that
and
Thus, we further deduce that
for
, where the last inequality holds since
. □
By Lemma 2, we only need to choose
and
It is derived in ([
13], cf. (16) and (17)) that the above error estimates can be valid by choosing
, and
. So, the suitable corrections in (
6) and (
7) can be written as follows:
and
The fully discrete scheme
with the two corrections (
23) and (
24) as the starting two steps is referred to as the BDF3, and can restore third-order accuracy in time, which will be proved in the next section.
4. Error Estimates of the BDF3
In this part, we present the error estimates for the modified scheme (
25). We first consider the error estimate for a homogenous problem with
and
. In the proof of error estimates, we may explicitly or implicitly employ the identity
and the
-stability of the orthogonal projection
when needed.
Theorem 2. Let u and be the solutions of (1) and (25), respectively. Assuming and . Then, for , we have Proof. It suffices to consider the bound of
in view of Theorem 1. From (
9) and (
10) in Lemma 1, we can obtain
Since
, in view of (
17) with Lemma 2 and letting
, we have
Note that
, one has
. So, we have
Putting the error estimates of the two parts,
and
, into (
26), we complete the proof of the theorem. □
Next, we state the error estimate for the homogenous problem with and .
Theorem 3. Let u and be the solutions of (1) and (25), respectively. Assuming and . Then, for we have Proof. Let
. In view of (
26), we have
where
. By the triangle inequality, the term
has the following error estimate:
For the first term
, using the result in (
19) and the criterion (
21), we have
For the second term
, we have
Applying the error estimates in Lemma B.1. of [
13] again, we derive that
Since
we obtain the error estimate for
:
Consequently, by choosing
, we obtain
For the second term
, similar to the derivation of
in Theorem 2, we can easily derive the following inequality:
where the first inequality is valid since
. This, together with the error estimate of
and Theorem 1, ends the proof. □
Now, we consider the inhomogenous problem, i.e., and .
Theorem 4. Let u and be the solutions of (1) and (25) with , respectively. Then, we have Proof. It follows from (
9) and (
10) in Lemma 1 that
where
From the derivation in Theorem 2, we can easily obtain the following error estimate of
:
Next, we consider the two terms
and
, respectively. Through the triangle inequality, we can obtain
Since
, using the criterion (
16), we have
. For
, the following inequality holds:
Analogous to the derivation of the error estimates of
and
for
in Theorem 3, we can obtain
. So, we choose
and get
For the last term
, by using the Taylor expansion of
at
, one has
. So
. It suffices to analyze the error estimates for source terms of the form
and
, respectively. Assume that
. Then the term
has the following form:
where
. By repeating the similar argument in the above error estimate for
, one can derive that
. When
, the corresponding semidiscrete Galerkin solution in (
9) can be written as follows:
where the operator
is given by
Furthermore, by (
13), we have
from which we can derive the representation of the fully discrete solution
below:
with
. In view of (
14), we get
Denote
with the Dirac delta function
at
, we can rewrite (
27) as
Since
one can derive that
We show that the above inequality is valid for
. Indeed, by using the Taylor series expansion of
at
, we have
This Taylor expansion also holds for
. One can easily obtain
and
Besides, there holds the following estimates:
and
with
. Thus, combining with Theorem 1, we complete the proof. □
6. Numerical Examples
Now, we perform the numerical examples to test the error estimates of BDF3 (
25) and BDF4 (
28). Let
and
. We consider the following three types of problem data:
- (a)
and ;
- (b)
and ;
- (c)
and .
Here, is denoted as the characteristic function of the set S.
We remark that the three types of data considered above cover all cases of error theories, and therefore this is sufficient for our numerical tests. Moreover, we focus on the temporal convergence rate since the spatial one is well understood. Since the analytical solution of equation with the above data is hard to obtain, we use reference solution instead and the reference solution is computed by BDF4 with fixing
and
. The
norm errors are measured by
. The numerical results are demonstrated in
Table 1,
Table 2,
Table 3,
Table 4,
Table 5 and
Table 6. From these tables, we can see that when no correction terms are added to BDF3 and BDF4, the temporal convergence accuracy is only first-order accuracy for both (see
Table 1,
Table 2 and
Table 3), while after adding suitable correction terms (cf.
Table 4,
Table 5 and
Table 6), we observe the theoretical temporal third-/fourth-order accuracy, which numerically verifies the importance of adding correction terms and the correctness of our constructed correction terms in BDF3/BDF4.
In addition, we also numerically compare the BDF3 and BDF4 with the BDF2 proposed in [
1]; see
Table 4,
Table 5 and
Table 6. The numerical results further indicate the correctness of our error estimates presented in Theorems 2–5 and reveal that our schemes greatly improve the temporal convergence accuracy of the BDF2.