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Parameter uniform numerical method for a system of two coupled singularly perturbed parabolic convection-diffusion equations
- Li-Bin Liu^{1},
- Guangqing Long^{1} and
- Yong Zhang^{2}Email author
https://doi.org/10.1186/s13662-018-1907-1
© The Author(s) 2018
- Received: 26 March 2018
- Accepted: 28 November 2018
- Published: 5 December 2018
Abstract
In this paper, we propose a numerical scheme for a system of two linear singularly perturbed parabolic convection-diffusion equations. The presented numerical scheme consists of a classical backward-Euler scheme on a uniform mesh for the time discretization and an upwind finite difference scheme on an arbitrary nonuniform mesh for the spatial discretization. Then, for the time semidiscretization scheme, an a priori and an a posteriori error estimations in the maximum norm are obtained. It should be pointed out that the a posteriori error bound is suitable to design an adaptive algorithm, which is used to generate an adaptive spatial grid. It is proved that the method converges uniformly in the discrete maximum norm with first-order time and spatial accuracy, respectively, for the fully discrete scheme. At last, some numerical results are given to validate the theoretical results.
Keywords
- Singularly perturbed
- Upwind finite difference scheme
- Adaptive grid method
MSC
- 65M06
- 65M12
1 Introduction
If \(A\equiv 0\), the above problem (1) becomes a system of singularly perturbed parabolic reaction-diffusion equations. For these problems, some robust convergence numerical approaches on layer-adapted meshes are available in the literature, e.g., Shishkina and Shishkin [1, 2], Gracia et al. [3–6], Franklin et al. [7].
If \(A\not \equiv 0\), the above problem (1) is said to be a system of singularly perturbed parabolic convection-diffusion equations. Furthermore, when A is a diagonal matrix, problem (1) is said to be a weakly coupled system (i.e., coupled only through their reaction terms). Otherwise, problem (1) is called to be a strongly coupled system. As far as we know, systems of convection-diffusion equations are more delicate to handle, especially for the time-dependent convection-diffusion systems. Consequently, some researchers paid attention to the layer adapted mesh methods for some weakly coupled singularly perturbed second-order ordinary differential equation systems of convection-diffusion type; see [8–12] and the references therein. Recently, in [13–15], the authors considered some special strongly coupled system of singularly perturbed convection-diffusion problems and constructed some corresponding layer-adapted mesh approaches. Very recently, Rao and Srivastava [16] constructed a parameter uniform numerical method for a weakly coupled linear system of singularly perturbed parabolic convection-diffusion equations. As far as we know, the layer-adapted mesh approach requires a priori information about the location and width of the boundary layer. Therefore, it is very necessary to provide an adaptive moving grid approach which only uses little or no a priori information to solve a coupled system of singularly perturbed convection-diffusion problems.
In order to serve this purpose, Linß[17] constructed an adaptive moving grid method to solve a strongly coupled system of singularly perturbed second-order two-point boundary problems. However, he did not give the optimal uniform convergence analysis. For this reason, Liu and Chen [18] also developed an adaptive moving grid approach for a weakly coupled singularly perturbed system. They not only constructed a simple mesh monitor function, but also proved the uniform convergence of the presented numerical method. Furthermore, Liu and Chen [19] constructed an adaptive moving grid method for a special strongly coupled system of two singularly perturbed convection-diffusion problems and obtained an a posteriori error estimation in maximum norm.
In this paper, we will use some techniques developed in [18] to devise a uniformly convergent numerical scheme for (1) under the restriction that A is a diagonal matrix, which is also first-order accurate both in time and space direction. It should be pointed out that our adaptive grid method need not require any a priori information about the location and width of the boundary layer. Moreover, the monitor function presented in this paper is similar to arc length function which is easy to design a mesh generation algorithm.
Notations: Throughout this paper we use C, sometimes subscripted, to denote a generic positive constant that is independent of all perturbation parameters \(\varepsilon _{i}\), \(i=1,2\), and mesh parameters N, M. It may take different values in different places.
For vector-valued functions \(\mathbf{v}= (v_{1}(t),v_{2}(t) ) ^{T}\), set \(|\mathbf{v}|= (|v_{1}(t)|,|v_{2}(t)| )^{T}\) and \(\|\mathbf{v}\|_{\infty }=\max \{\|v_{1}\|_{\infty },\|v_{2}\| _{\infty } \}\).
A mesh function \(\varphi := \{\varphi (t_{i}) \}_{i=0}^{N}\) is a real-valued function. Define the discrete maximum norm for such functions by \(\|\varphi \|_{\infty }=\max_{i=0,1,\ldots ,N}| \varphi (t_{i})|\). For vector mesh functions \(\mathbf{V}:= \{(V _{1}(t_{i}),V_{2}(t_{i}))^{T} \}_{i=0}^{N}\), we define \(\|\mathbf{V}\|_{\infty }=\max \{\|V_{1}\|_{\infty }, \|V_{2}\| _{\infty } \}\).
2 The time semidiscretization
In this section, we mainly construct the time semidiscretization scheme which is important for the convergence analysis of the fully discrete scheme.
2.1 The semidiscrete scheme in time
2.2 Convergence analysis
In the following, to analyze the uniform convergence of the solution \(\mathbf{u}^{n}(x)\) of (4) to the exact solution \(\mathbf{u}(x,t _{n})\) of (1), the maximum principle for equations (4) is established. Then, using this principle, a stability result is derived.
Theorem 2.1
(Maximum principle)
Assume that \((I+\Delta t L_{x,\varepsilon }) \mathbf{u}^{n+1}(x)\geq \mathbf{0}\) in Ω and \(\mathbf{u}^{n+1}(0) \geq \mathbf{0}\), \(\mathbf{u}^{n+1}(1)\geq \mathbf{0}\), then \(\mathbf{u}^{n+1}(x)\geq \mathbf{0}\) in Ω.
Proof
Let \(u_{1}^{n+1}(p)=\min_{x\in [0,1]}u_{1}^{n+1}(x)\) and \(u_{2}^{n+1}(q)=\min_{x\in [0,1]}u_{2}^{n+1}(x)\). Assume without loss of generality that \(u_{1}^{n+1}(p)\leq u_{2}^{n+1}(q)\) if the result of this theorem is not true. In other words, \(u_{1}^{n+1}(p)<0\).
It follows from the above Theorem 2.1 that we can obtain the following Lemma 2.1 (one can see the proof of Lemma 2.2 in [20]).
Lemma 2.1
Lemma 2.2
Proof
The proof is similar to Lemma 3 of [16]. □
Lemma 2.3
Proof
Finally, based on the above Theorem 2.1, Lemmas 2.1, 2.2, and 2.3, we can obtain the following convergence result.
Theorem 2.2
Thus, scheme (4) is uniformly convergent of first order.
3 The spatial discretization and adaptive spatial grid algorithm
3.1 The upwind finite difference scheme
3.2 Adaptive spatial grid algorithm
Therefore, to obtain an adaptive equidistribution grid and the corresponding numerical solution, we construct the following iteration algorithm:
Step 1. Let \(n=1\).
Step 2. For \(k=0\), let \(\{x_{i}^{n,(k)}=i/N,i=0,1,\ldots ,N \}\) be the initial uniform spatial mesh for \(n=1\), otherwise, choose \(\{x_{i}^{n-1}\}\) for the initial mesh.
Step 6. Let \(k=k+1\) and go to Step 3.
Step 7. Choose \(\{x_{i}^{n,k-1} \}\) as the final mesh \(\{x_{i}^{n} \}\) and let \(U_{j,i}^{n,k-1}=U_{j,i}^{n}\).
Step 8. Let \(n=n+1\), return to Step 2.
Step 9. If \(n=M\), go to Step 10.
Step 10. Stop. \(U_{j,i}^{n}\) is the numerical solution, grid for each time level given by \(\{x_{i}^{n}\}\).
4 Error analysis
4.1 Preliminary results
Lemma 4.1
Corollary 4.1
4.2 A priori error analysis
Based on Corollary 4.1, we get the following a priori error estimation.
Lemma 4.2
Proof
Lemma 4.3
Proof
Using the results given in Lemmas 3–4 of [8], we can obtain the desired results. □
Based on the above Lemmas 4.2 and 4.3, we can obtain the following convergence result.
Theorem 4.1
Proof
Finally, the desired estimate (27) follows from Lemma 4.2, (30) and (31). □
Remark 1
Theorem 4.1 gives the optimal first-order convergence of the upwind finite difference scheme uniformly in the perturbation parameters \(\varepsilon _{i}\) (\(i=1,2\)). However, it is hard to obtain the adaptive mesh \(\{x_{i}^{n+1} \}_{i=0}^{N}\) by using mesh equidistribution formula (31) since this requires the prior information about the exact solution. Thus, in practical computation, we often utilize an a posteriori error estimation and the corresponding monitor function to design a mesh generation algorithm.
4.3 A posteriori error analysis
Based on the above Lemma 4.1, we can obtain an a posteriori error estimation for the numerical scheme (20) at time level \(t_{n+1}=(n+1)\Delta t\).
Theorem 4.2
Proof
Furthermore, similar to Theorem 4.1 of [18], we obtain the following result.
Theorem 4.3
At last, we give the ε-uniform convergence of the fully discrete scheme (11) on the adaptive mesh produced by equidistributing the monitor function (16).
Theorem 4.4
Proof
The proof is similar to Theorem 4.7 of [27]. □
5 Numerical examples and discussion
In this section, we show the numerical results of two examples to verify the theoretical results. For comparison purposes, we use the presented full discretization scheme (11) on the piecewise-uniform Shishkin mesh, which is constructed as follows.
Example 5.1
Maximum error \(E_{\varepsilon _{1},\varepsilon _{2}}^{N,M}\) for Example 5.1 using the adaptive grid method with \(\varepsilon _{1}=10^{-2}\)
\(\varepsilon _{2}\) | Number of intervals N/time size Δt | ||||
---|---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | \(512/\frac{1}{320}\) | |
10^{−2} | 1.4250e − 01 | 7.9154e − 02 | 4.0946e − 02 | 1.9700e − 02 | 1.0618e − 02 |
10^{−3} | 1.1938e − 01 | 7.1365e − 02 | 4.0316e − 02 | 2.1692e − 02 | 1.1134e − 02 |
10^{−4} | 1.6654e − 01 | 9.3826e − 02 | 5.4498e − 02 | 2.7184e − 02 | 1.3513e − 02 |
10^{−5} | 1.8941e − 01 | 1.1599e − 01 | 6.5234e − 02 | 3.2013e − 02 | 1.7915e − 02 |
10^{−6} | 2.1617e − 01 | 1.4118e − 01 | 7.4591e − 02 | 3.9796e − 02 | 2.0694e − 02 |
10^{−7} | 2.4067e − 01 | 1.3939e − 01 | 7.0974e − 02 | 4.2329e − 02 | 2.3750e − 02 |
Rate of convergence \(r_{\varepsilon _{1},\varepsilon _{2}} ^{N,M}\) for Example 5.1 using the adaptive grid method with \(\varepsilon _{1}=10^{-2}\)
\(\varepsilon _{2}\) | Number of intervals N/time size Δt | |||
---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | |
10^{−2} | 0.8482 | 0.9509 | 1.0555 | 0.8917 |
10^{−3} | 0.7423 | 0.8239 | 0.8942 | 0.9622 |
10^{−4} | 0.8278 | 0.7838 | 1.0034 | 1.0084 |
10^{−5} | 0.7075 | 0.8303 | 1.0270 | 0.8375 |
10^{−6} | 0.6146 | 0.9205 | 0.9064 | 0.9434 |
10^{−7} | 0.7879 | 0.9738 | 0.7456 | 0.8337 |
Maximum error \(E_{\varepsilon }^{N,M}\) for Example 5.1 using the adaptive grid method with \(\varepsilon _{2}=10^{-2}\)
\(\varepsilon _{1}\) | Number of intervals N/time size Δt | ||||
---|---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | \(512/\frac{1}{320}\) | |
10^{−3} | 1.5659e − 01 | 9.0728e − 02 | 4.9472e − 02 | 2.5685e − 02 | 1.3157e − 02 |
10^{−4} | 1.6708e − 01 | 9.6715e − 02 | 5.2283e − 02 | 2.7677e − 02 | 1.4250e − 02 |
10^{−5} | 1.8039e − 01 | 9.7457e − 02 | 5.2987e − 02 | 2.8267e − 02 | 1.5248e − 02 |
10^{−6} | 1.7981e − 01 | 9.7493e − 02 | 5.3310e − 02 | 2.8267e − 02 | 1.6714e − 02 |
10^{−7} | 1.8023e − 01 | 1.0099e − 01 | 5.3679e − 02 | 2.8865e − 02 | 1.6969e − 02 |
Rate of convergence \(r_{\varepsilon _{1},\varepsilon _{2}} ^{N,M}\) for Example 5.1 using the adaptive grid method with \(\varepsilon _{2}=10^{-2}\)
\(\varepsilon _{1}\) | Number of intervals N/time size Δt | |||
---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{60}\) | \(256/\frac{1}{160}\) | |
10^{−3} | 0.7874 | 0.8749 | 0.9457 | 0.9651 |
10^{−4} | 0.7887 | 0.8874 | 0.9177 | 0.9577 |
10^{−5} | 0.8883 | 0.8791 | 0.9065 | 0.8905 |
10^{−6} | 0.8831 | 0.8709 | 0.9153 | 0.7581 |
10^{−7} | 0.8356 | 0.9118 | 0.8950 | 0.7664 |
Comparison of numerical results of Example 5.1 with Shishkin mesh with \(\varepsilon _{2}=10^{-2}\)
N/Δt | \(\varepsilon _{1}=10^{-4}\) | \(\varepsilon _{1}=10^{-6}\) | ||
---|---|---|---|---|
Shishkin mesh | Adaptive mesh | Shishkin mesh | Adaptive mesh | |
\(32/\frac{1}{20}\) | 4.8924e − 01 | 1.6708e − 01 | 4.9039e − 01 | 1.7981e − 01 |
0.0917 | 0.7887 | 0.0920 | 0.8831 | |
\(64/\frac{1}{40}\) | 4.5911e − 01 | 9.6715e − 02 | 4.6011e − 01 | 9.7493e − 02 |
0.0354 | 0.8874 | 0.0357 | 0.8709 | |
\(128/\frac{1}{80}\) | 4.4799e − 01 | 5.2283e − 02 | 4.4885e − 01 | 5.3310e − 02 |
0.0151 | 0.9177 | 0.0154 | 0.9153 | |
\(256/\frac{1}{160}\) | 4.4334e − 01 | 2.7677e − 02 | 4.4409e − 01 | 2.8267e − 02 |
Example 5.2
Maximum error \(E_{\varepsilon }^{N,M}\) for Example 5.2 using the adaptive grid method with \(\varepsilon _{1}=10^{-2}\)
\(\varepsilon _{2}\) | Number of intervals N/time size Δt | ||||
---|---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | \(512/\frac{1}{320}\) | |
10^{−2} | 7.5287e − 02 | 4.7758e − 02 | 2.7686e − 02 | 1.5234e − 02 | 7.9016e − 03 |
10^{−3} | 8.0020e − 02 | 4.9508e − 02 | 2.9305e − 02 | 1.6339e − 02 | 8.6897e − 03 |
10^{−4} | 1.1255e − 01 | 6.8986e − 02 | 3.9675e − 02 | 2.1337e − 02 | 1.0733e − 02 |
10^{−5} | 1.2378e − 01 | 8.0042e − 02 | 4.5942e − 02 | 2.6356e − 02 | 1.5532e − 02 |
10^{−6} | 1.2592e − 01 | 8.2114e − 02 | 4.8879e − 02 | 2.7391e − 02 | 1.5791e − 02 |
10^{−7} | 1.2119e − 01 | 7.6072e − 02 | 4.8654e − 02 | 2.8021e − 02 | 1.7446e − 02 |
Rate of convergence \(r_{\varepsilon _{1},\varepsilon _{2}} ^{N,M}\) for Example 5.2 using the adaptive grid method with \(\varepsilon _{1}=10^{-2}\)
\(\varepsilon _{2}\) | Number of intervals N/time size Δt | |||
---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | |
10^{−2} | 0.6567 | 0.7866 | 0.8619 | 0.9471 |
10^{−3} | 0.6927 | 0.7566 | 0.8428 | 0.9109 |
10^{−4} | 0.7062 | 0.7981 | 0.8949 | 0.9913 |
10^{−5} | 0.6289 | 0.8009 | 0.8017 | 0.7629 |
10^{−6} | 0.6168 | 0.7484 | 0.8355 | 0.7946 |
10^{−7} | 0.6718 | 0.6448 | 0.7960 | 0.6836 |
Maximum error \(E_{\varepsilon }^{N,M}\) for Example 5.2 using the adaptive grid method with \(\varepsilon _{2}=10^{-2}\)
\(\varepsilon _{1}\) | Number of intervals N/time size Δt | ||||
---|---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | \(512/\frac{1}{320}\) | |
10^{−3} | 8.5765e − 02 | 5.4978e − 02 | 3.2185e − 02 | 1.7851e − 02 | 9.4792e − 03 |
10^{−4} | 1.0625e − 01 | 6.7554e − 02 | 3.9103e − 02 | 2.1170e − 02 | 1.0681e − 02 |
10^{−5} | 1.1804e − 01 | 7.8694e − 02 | 4.5355e − 02 | 2.6197e − 02 | 1.4536e − 02 |
10^{−6} | 1.1955e − 01 | 8.0068e − 02 | 4.7762e − 02 | 2.7255e − 02 | 1.5683e − 02 |
10^{−7} | 1.1639e − 01 | 7.4345e − 02 | 4.9340e − 02 | 2.8595e − 02 | 1.7408e − 02 |
Rate of convergence \(r_{\varepsilon _{1},\varepsilon _{2}} ^{N,M}\) for Example 5.2 using the adaptive grid method with \(\varepsilon _{2}=10^{-2}\)
\(\varepsilon _{1}\) | Number of intervals N/time size Δt | |||
---|---|---|---|---|
\(32/\frac{1}{20}\) | \(64/\frac{1}{40}\) | \(128/\frac{1}{80}\) | \(256/\frac{1}{160}\) | |
10^{−3} | 0.6415 | 0.7725 | 0.8504 | 0.9132 |
10^{−4} | 0.6533 | 0.7888 | 0.8853 | 0.9870 |
10^{−5} | 0.5850 | 0.7950 | 0.7919 | 0.8498 |
10^{−6} | 0.5783 | 0.7454 | 0.8093 | 0.7973 |
10^{−7} | 0.6467 | 0.5915 | 0.7870 | 0.7360 |
Comparison of numerical results of Example 5.2 with Shishkin mesh with \(\varepsilon _{2}=10^{-2}\)
N/Δt | \(\varepsilon _{1}=10^{-4}\) | \(\varepsilon _{1}=10^{-6}\) | ||
---|---|---|---|---|
Shishkin mesh | Adaptive mesh | Shishkin mesh | Adaptive mesh | |
\(32/\frac{1}{20}\) | 9.1770e − 02 | 1.0625e − 01 | 9.2836e − 02 | 1.7981e − 01 |
0.5076 | 0.6533 | 0.4936 | 0.5783 | |
\(64/\frac{1}{40}\) | 6.4546e − 02 | 6.7754e − 02 | 6.5938e − 02 | 9.7493e − 02 |
0.6481 | 0.7888 | 0.6587 | 0.7454 | |
\(128/\frac{1}{80}\) | 4.1187e − 02 | 3.9103e − 02 | 4.1769e − 02 | 5.3310e − 02 |
0.7238 | 0.8853 | 0.7118 | 0.8093 | |
\(256/\frac{1}{160}\) | 2.4938e − 02 | 2.1170e − 02 | 2.5503e − 02 | 2.8267e − 02 |
6 Concluding
In this paper, we have discussed an adaptive grid method for a coupled system of two singularly perturbed parabolic convection-diffusion equations. We first apply the backward-Euler scheme to discretize problem (1) with respect to time derivative and the upwind finite difference scheme on an arbitrary nonuniform mesh to approximate the spatial derivative. Then, at each time level \(t_{n}=n \Delta t\), a positive monitor function given in (16) is used to design an adaptive grid generation algorithm. An a posteriori error estimate for the proposed numerical scheme is obtained. We also establish that the presented adaptive grid approach is of first-order rate of convergence in both the spatial and temporal variables. Finally, some numerical results are conducted to support the theoretical results and also, to demonstrate the effectiveness of the adaptive spatial grid obtained by the above mesh generation algorithm.
Declarations
Acknowledgements
The authors would like to express their sincere gratitude to the editor and referees, who gave us valuable comments to improve this paper. The authors also thanks “BAGUI Scholar” Program of Guangxi Zhuang Autonomous Region of China for its support.
Funding
This work is supported by the National Science Foundation of China (11761015, 11826211, 11826212, 11461011), the Natural Science Foundation of Guangxi (2017GXNSFBA198183), the key project of Guangxi Natural Science Foundation (2017GXNSFDA198014), the key project of Anhui natural science research (KJ2015A213), the projects of Excellent Young Talents Fund in Universities of Anhui Province (gxyq2017105), the Anhui Provincial Natural Science Foundation (1708085QA13).
Authors’ contributions
LBL, GQL, and YZ carried out the numerical method and numerical experiments, drafted the manuscript, and participated in the design of the study and performed proof of results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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