Abstract
In this paper, we deal with a singularly perturbed parabolic convection-diffusion problem. Shishkin mesh and a hybrid third-order finite difference scheme are adopted for the spatial discretization. Uniform mesh and the backward Euler scheme are used for the temporal discretization. Furthermore, a preconditioning approach is also used to ensure uniform convergence. Numerical experiments show that the method is first-order accuracy in time and almost third-order accuracy in space.
1. Introduction
We consider the singularly perturbed parabolic problem posed on the domain as follows:where , , and is a small positive perturbation parameter satisfyingwith a positive constant , and is the initial value when . We assume that the functions , , and are sufficiently smooth, and that and satisfywhere is a positive constant.
With these conditions, there exists a unique -solution of problem (1) (see, for instance, [1]). The -solutions of problem (1) satisfied from its original function to its fourth partial derivative for spatial variable are continuous, and solutions from its original function to its second partial derivative for temporal variable are also continuous.
Singular perturbation problems play an important role in many areas, such as astronomy, mechanics, and fluid dynamics. It also has a broad background and important applications in control systems with different time scales [2–4]. It is especially important to find a uniform and effective approximate solution when the exact solution cannot be obtained. There are many methods for solving singular perturbation problems. Recent convergence analysis of the finite element method is referred to [5–14]. Except the finite element method, the finite difference method is the most widely used one at present. Nowadays, more and more people begin to study higher-order finite difference schemes for solving singular perturbation problems. In 1988, Vulanović in [15] proposed a third-order hybrid finite difference scheme and showed numerical results on the Shishkin mesh. Afterwards, Vulanovic and Nhan in [16] improved on what had already been done and proposed a new uniformly convergent numerical scheme. Both the methods proposed in [15, 17] have been analyzed on a piecewise-uniform Shishkin mesh and were proved to be almost third-order accuracy in space. Comparing the previous scheme, the difference is that when is large enough, the accuracy of the new scheme is better than the that of the previous scheme. However, when is small enough, there is no difference between these two methods.
In this paper, our primary aim is to propose and analyze a higher-order hybrid finite difference scheme for problem (1). This is accomplished by discretizing the domain using the Shishkin mesh and by considering the uniform mesh in the temporal direction. In order to obtain the fully discrete scheme, we adopt the two-stage discretization process. The first stage consists of discretizing the time derivative with the backward difference scheme on the uniform mesh. In the second stage, discretize in the spatial direction by utilizing a hybrid finite difference scheme on the Shishkin mesh.
The ultimate goal of numerical methods for problem (1) is to obtain a series of discrete solutions so as to achieve a numerical approximation of the continuous solution. Such that its error converges to uniformly as , where is the number of discretization on the spatial mesh. Apart from this, in numerical experiments, we also need to illustrate that the proposed scheme is almost third-order accurate in space.
The rest of the paper is as follows. In Section 2, we define the meshes for temporal and spatial discretization and introduce some special difference operators. In Section 3, we define some difference operators and the final finite difference scheme. In Section 4, we give the linear equations needed to solve the problem and get the coefficient matrix and the right end term. In Section 5, we preprocess the coefficient matrix. In Section 6, we give the pseudo code to solve the problem. In Section 7, we give the results of numerical experiments. In Section 8, some final conclusions are given.
2. The Mesh
Here, in this section, we describe the uniform mesh for the temporal discretization of the domain and the Shishkin mesh for the spatial discretization of the domain .
We will often use the assumption thatwhere is a sufficiently small positive constant which is independent of both and . All constants, independent of and , are denoted generically by .
2.1. The Uniform Mesh
For the time domain , we use a uniform mesh with time step , such thatwhere is the number of mesh points in the -direction on the interval .
2.2. Shishkin Mesh
Since problem (1) has a boundary layer along the side , the mesh should be condensing in the neighborhood of . Definewith (the proof of range of was given in [16]). To define the piecewise-uniform mesh, we divide the domain into two subdomains, such that and then divide each of the subdomains into equal intervals. Set
Now, we denote the spatial grids bywhereand be a positive even integer. Here, the transition point separates the coarse and fine portions of the mesh.
Moreover, definewhere is some fixed constant.
3. Discretization
In this section, we will give different difference operators corresponding to different points on the Shishkin mesh and combine these difference operators to form the final numerical scheme.
Firstly, we denote by the approximation of at point and set . Then, we use , , and as the approximations of , , and , respectively. They are defined by the following equation [16]:where is the step size of a uniform mesh and is a constant satisfying . We set as the truncation error between the numerical solution and the exact solution.
Firstly, .
Lemma 1. Suppose that
The truncation error associated to satisfieswhere is the step size.
Proof. We substitute for in operator and apply the Taylor expansion to obtainThus,Secondly, .
Lemma 2. Suppose that
The truncation error associated to satisfieswhere is the step size.
Proof. Similar to above, we substitute for in operator and again apply the Taylor expansion to obtainThus,In conclusion, both and are third-order accurate with respect to the spatial variable for any value of ; if , is transformed into the classical three-point scheme. And in the same way, operator (27), (29), (30), and time difference operator can all be proven.
Moreover, .
Lemma 3. Assume that
The truncation error associated to satisfies that if ,elsewhere is the step size.
Proof. Once more, substituting for in operator and applying the Taylor expansion results inwe can found that the operator in general is second-order accurate, and if , it is third-order accurate.
These schemes can be used to create the following difference operator :where and . The operator is only used as part of the discretization on the Shishkin grid because the Shishkin grid is not uniform in the entire computational domain. More specifically, the difference operator cannot be applied at and .
Remark 1. Because scheme has point in it, the Shishkin grid used is divided into two intervals, and . Since the step size is different and the point spans two intervals, the difference operator cannot be applied at either or .
Now, we introduce the difference operator as follows:withwhere and , and it is also third-order accurate.
Then, we give the scheme at point by means of one-side difference schemes as follows:withandwhere and , and both and have third-order accuracy in space.
In addition, about the -direction, the discretization of by the backward Euler scheme is defined bywith the time step , and it is first-order accurate with respect to the temporal variable .
Finally, we combine with three difference operators , , and at different points, respectively, and finally propose the following numerical scheme:with .
4. Linear Problem
The corresponding difference schemes of , , and at point and at point are substituted into equation (1). When combined with scheme (32), the following linear equations (33) are obtained:where is the approximation of and , and are defined as follows: if ,where is the time steps, is the space steps defined by (7), , and is defined by (11). If ,where is the time steps, is the space steps defined by (7), , and is defined by (11). If ,where is the time steps, is the space steps defined by (7), , and is defined by (11). If ,where is the time steps, is the space steps defined by (7), , and is defined by (11).
Finally, the linear system for numerical scheme (32) is obtained, that is,
Here, the coefficient matrix is defined bywhere each unwritten element is 0. The unknown term is defined by with and , and the right end term is defined by .
5. Preconditioning
In this section, we analyze the matrix , which corresponds to the first scheme (32) and is acquired in Section 4. We need to assume (4) and that is sufficiently large andwhere is a positive integer independent of . There exist constants and such that (4) and (40) are satisfied, and the matrix has the following structure:where each unwritten element is 0.
Multiply its equations by . We do this to achieve consistency uniform in , but at the same time, the coefficient matrix gets preconditioned as well (the preconditioning be described in [18, 19]).
Thus, we take the matrix of the preconditioned system as follows:where
6. Pseudo Code
In this section, the pseudo code needed to solve problem (1) using numerical scheme (32) in MATLAB will be presented. In general, if mathematical tools are used to solve problem like this, by scheme (32), there are six steps as follows [17]:(1)Set the uniform mesh for temporal variable ( is the total number of points in -direction)(2)Set the Shishkin mesh for spatial variable ( is the total number of points in space)(3)Write down the coefficient matrix and the right end term for the linear system as follows: where is the approximation of , is the space steps, are the elements of the coefficient matrix, and is the right end term (it is defined in Section 4)(4)A new matrix (from Section 5) is obtained by preprocessing the matrix (5)The new matrix and the right end term are used to solve problem (1)(6)The maximum pointwise errors and the orders of convergence are calculated
7. Numerical Experiments
In this section, we shall present the numerical results obtained by the proposed numerical schemes (32) for the test problem (45) on the piecewise-uniform rectangular mesh . In both cases, we perform the numerical experiments by choosing the constants and in (6) and the time step .
For numerical tests, we consider the following singularly perturbed parabolic problem:where . We choose the initial data and the exact solution for problem (45) as follows [16]:
As the exact solution of problem (45) is known, we calculate the maximum pointwise error byfor each , where and denote the exact solution and numerical solution on , respectively. The convergence order is calculated by the following formula:
The maximum pointwise errors and the order of convergence by using schemes (32) are presented in Table 1. In table, we can observe the -uniform convergence of the numerical scheme. The order of convergence in Table 1 is first-order due to the effect of time error. In order to justify the spatial order of convergence precisely, we take and the order of convergence is defined by
The numerical results are presented in Table 2, where the spatial convergence order is almost third-order.
8. Conclusion
A hybrid scheme is proposed for obtaining a numerical solution to the singularly perturbed parabolic problem. The idea is based on the methods presented in the existing research study [15–19]. It can be seen from the results of numerical experiments, whether in space or in time, the scheme is robust insomuch the error of the numerical solution does not increase when . On the contrary, the proposed schemes improve as diminishes, becoming almost third-order accurate with the spatial variable and first-order accurate with the temporal variable. The numerical results were compared with those from literature [15–18, 20, 21] which showed that all results reach the expected order of convergence. However, so far it is not possible to construct an arbitrary high-order difference scheme for Shishkin grids, meaning further research is needed. It should be noted that parallelization is not discussed in this article, so the reader is encouraged to refer to additional work [20, 21].
Data Availability
The data used to support the findings of this study are openly available in web of science at https://doi.org/10.1016/j.amc.2020.125495.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was partially supported by National Natural Science Foundation of China (11771257 and 11601251).