## 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 *x*.
For any continuous function

## 2 The Continuous Problem

Consider the initial-boundary value problem

(2.1)

The initial condition ϕ is also smooth on *f* is smooth on *Caputo fractional derivative*
which is defined [1] by

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 *y* whose derivatives satisfy certain bounds. Transforming back to the original problem (2.1), under certain conditions on its data one obtains the following bounds on the derivatives of *u*:

(2.3)

for all

In [10, Theorem 2.1] the estimates in (2.3) are proved assuming that

for all *t* and

where

## 3 The Discrete Problem

The solution of problem (2.1) is approximated by the solution of a finite difference scheme on a mesh *M* and *N* be positive integers. Set *u* computed at the mesh point

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 *N* satisfies

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 *k*th time cell

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, *k*. Using this inequality and the inductive hypothesis, we require the inequality

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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**Received:**2017-1-27

**Revised:**2017-5-27

**Accepted:**2017-6-6

**Published Online:**2017-7-6

**Published in Print:**2018-1-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston