Abstract
A standard finite difference method on a uniform mesh is used to solve a time-fractional convection-diffusion initial-boundary value problem. Such problems typically exhibit a mild singularity at the initial time
1 Introduction
In this paper, we examine the convergence rate of numerical approximations to a time-fractional convection-diffusion problem using a
standard finite difference method on a uniform mesh. Initial-boundary value problems of this type, where the time derivative is fractional, have solutions that are mildly singular at the initial time
This paper is a companion paper to [10], where it was shown that the convergence rate of the same finite difference scheme on a uniform mesh was
Our analysis is carried out in the discrete
Notation.
In this paper C denotes a generic constant that depends on the data of the boundary value problem (2.1) but is independent of T and of any mesh used to solve (2.1) numerically. Note that C can take different values in different places. For all
2 The Continuous Problem
Consider the initial-boundary value problem
(2.1)
The initial condition ϕ is also smooth on
where
is the Riemann–Liouville fractional integral operator of order
There is no loss of generality in assuming homogeneous boundary conditions in (2.1b), because inhomogeneous boundary conditions are easily made homogeneous by a simple change of variable.
Under the transformation
problem (2.1) becomes
(2.2)
Note that no first-order derivative in space appears in (2.2a), and (2.1d) implies that
(2.3)
for all
In [10, Theorem 2.1] the estimates in (2.3) are proved assuming that
for all
where
3 The Discrete Problem
The solution of problem (2.1) is approximated by the solution of a finite difference scheme on a mesh
The first and second-order spatial derivatives are discretised using standard approximations:
The Caputo fractional derivative
is approximated by the classical L1 approximation
(3.1)
Thus, we approximate (2.1) by the discrete problem
(3.2)
This discretisation of (2.1) is standard.
To ensure the stability of the discrete operator
After some minor modifications in the proof of [10, Theorem 5.2] to handle the term
for some constant C. In particular, the method has the low order of convergence
4 Error Analysis
The structure of our error analysis is the standard finite difference technique of estimating the truncation error at each mesh point, then invoking a stability argument to derive an error bound for the computed solution
The estimate of the truncation error in space is standard: using (2.3a), one gets
(4.1)
The truncation error in time is more tricky to estimate and this is done in the next lemma.
Lemma 1.
Assume that u satisfies (2.3). Then there exists a positive constant C such that for each mesh point
Proof.
We modify the argument of [10, Lemma 5.1]. By (3.1a) and the definition of
where for
The following four bounds are established in [10, equations (5.9), (5.10), (5.11) and (5.14)]:
(4.4)
It remains to bound
where
For
Hence
by the Mean Value Theorem. Combine this bound with (4.4) to complete the proof. ∎
Observe that
Next, we derive some new information about the stability constants that appear in [10, Section 4].
It follows from [10, Lemma 4.2] that the computed solution
for
Note that when the mesh is uniform, the weights
Lemma 2.
The coefficients
Proof.
First,
which is established in [5, Lemma 3.2]. ∎
The next result, which is a variant of [10, Lemma 4.3], bounds a weighted sum of the
Lemma 3.
Let the parameter β satisfy
Proof.
By Lemma 2, we have
But for
Hence,
by [1, Theorem D.6].
Substituting this inequality into (4.7) and using
This completes the proof. ∎
We can now prove our main result.
Theorem 4.
Assume that u satisfies (2.3). Then, for
for some constant C.
Proof.
Fix
By (4.5) we then obtain
Invoking Lemma 3 (with
The bound in (4.8) implies that for any fixed
That is, on any subdomain that is bounded away from
5 Numerical Results
In this section we give numerical results for the numerical method (3.2) applied to two particular examples from the problem class (2.1). In the first example the exact solution of the problem is known; in the second example it is unknown, so we estimate the order of convergence using the double-mesh principle [2]. In these numerical experiments we always take
Example 5.1.
Consider the constant coefficient homogeneous problem
with initial condition
where
In Figure 1 we display the computed solutions with scheme (3.2) for
For Example 5.1 we computed the maximum errors
and the orders of convergence
where
Considering the convergence in time, identified by the factor
α | |||||
0.4 | 8.438E-2 | 6.714E-2 | 5.282E-2 | 4.120E-2 | 3.191E-2 |
0.330 | 0.346 | 0.359 | 0.368 | ||
0.6 | 3.759E-2 | 2.512E-2 | 1.672E-2 | 1.109E-2 | 7.342E-3 |
0.581 | 0.588 | 0.592 | 0.595 | ||
0.8 | 1.121E-2 | 6.401E-3 | 3.666E-3 | 2.102E-3 | 1.206E-3 |
0.809 | 0.804 | 0.803 | 0.802 |
α | |||||
0.4 | 1.024E-2 | 4.966E-3 | 2.436E-3 | 1.214E-3 | 6.050E-4 |
1.044 | 1.027 | 1.005 | 1.005 | ||
0.6 | 1.300E-2 | 6.432E-3 | 3.190E-3 | 1.595E-3 | 7.965E-4 |
1.015 | 1.012 | 1.000 | 1.002 | ||
0.8 | 9.844E-3 | 5.123E-3 | 2.644E-3 | 1.361E-3 | 6.963E-4 |
0.942 | 0.954 | 0.959 | 0.966 |
Example 5.2.
Consider the variable coefficient inhomogeneous problem
(5.1)
Figure 3 displays the computed solution for
The exact solution of Example 5.2 is unknown and we shall estimate the order of convergence using the two-mesh principle [2]. Let
and hence the estimated orders of convergence
Tables 3 and 4 give the maximum two-mesh differences
and their corresponding orders of convergence for Example 5.2 in the domain
α | |||||
0.4 | 1.031E-2 | 8.673E-3 | 7.123E-3 | 5.740E-3 | 4.558E-3 |
0.250 | 0.284 | 0.311 | 0.333 | ||
0.6 | 4.935E-3 | 3.338E-3 | 2.234E-3 | 1.486E-3 | 9.857E-4 |
0.564 | 0.579 | 0.588 | 0.593 | ||
0.8 | 1.661E-3 | 9.441E-4 | 5.368E-4 | 3.060E-4 | 1.748E-4 |
0.815 | 0.815 | 0.811 | 0.808 |
α | |||||
0.4 | 5.849E-4 | 2.783E-4 | 1.351E-4 | 6.711E-5 | 3.337E-5 |
1.072 | 1.042 | 1.010 | 1.008 | ||
0.6 | 1.148E-3 | 5.457E-4 | 2.628E-4 | 1.291E-4 | 6.356E-5 |
1.073 | 1.054 | 1.025 | 1.023 | ||
0.8 | 1.335E-3 | 6.752E-4 | 3.387E-4 | 1.703E-4 | 8.531E-5 |
0.984 | 0.995 | 0.992 | 0.997 |
In [10], numerical results were given for the particular case of a fractional reaction-diffusion equation (i.e., with
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 91430216
Award Identifier / Grant number: NSAF U1530401
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2016-75139-R
Funding statement: The research of José Luis Gracia was partly supported by the Institute of Mathematics and Applications (IUMA), the project MTM2016-75139-R funded by the Spanish Ministry of Economy and Competitiveness (MINECO) and the Diputación General de Aragón. The research of Martin Stynes was supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF U1530401.
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