| Citation: |
Mohammad Prawesh Alam, Geetan Manchanda, Arshad Khan. AN ε-UNIFORMLY CONVERGENT METHOD FOR SINGULARLY PERTURBED PARABOLIC PROBLEMS EXHIBITING BOUNDARY LAYERS[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 2089-2120. doi: 10.11948/20220382
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AN ε-UNIFORMLY CONVERGENT METHOD FOR SINGULARLY PERTURBED PARABOLIC PROBLEMS EXHIBITING BOUNDARY LAYERS
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Abstract
A numerical method is proposed for singularly perturbed parabolic convection-diffusion equation whose solution exhibits boundary layers near the right endpoints of the domain of consideration. The method encompasses the Crank-Nicolson scheme on a uniform mesh in temporal direction and quartic B-spline collocation method on piecewise-uniform (i.e., Shishkin mesh) mesh in space directions, respectively. Through rigorous convergence analysis, the method has shown theoretically fourth-order convergent in space direction and second-order convergent in the time direction. We have solved two numerical examples to prove the efficiency and robustness of the method and to validate the theoretical results. Since the exact/analytical solution to the problem is not known, hence we applied the double mesh principle to compute the maximum absolute errors. Additionally, some numerical simulations are displayed to produce the conclusiveness of determining layer behaviour and their locations.
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References
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Mohammad Prawesh Alam, Geetan Manchanda, Arshad Khan. AN ε-UNIFORMLY CONVERGENT METHOD FOR SINGULARLY PERTURBED PARABOLIC PROBLEMS EXHIBITING BOUNDARY LAYERS[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 2089-2120. doi: 10.11948/20220382
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Figure 1.
Approximate solutions of the Problem 6.1 for
$ n = 128 $ $ \varepsilon= 2^{-10} $ -
Figure 2.
Approximate solutions of the Problem 6.1 for
$ n = 128 $ $ \varepsilon= 2^{-20} $ -
Figure 3.
Loglog plot of errors the Problem 6.1 for
$ n = 128 $ $ \varepsilon= 2^{-10} $ -
Figure 4.
Loglog plot of errors of the Problem 6.1 for
$ n = 128 $ $ \varepsilon= 2^{-20} $ -
Figure 5.
Approximate solutions of the Problem 6.1 at various time level with
$ n = 64 $ $ \varepsilon= 2^{-20} $ -
Figure 6.
Approximate solutions of the Problem 6.1 at various time level with
$ n =128 $ $ \varepsilon= 2^{-20} $ -
Figure 7.
Approximate solutions of the Problem 6.2 for
$ n = 128 $ $ \varepsilon= 2^{-10} $ -
Figure 8.
Approximate solutions of the Problem 6.2 for
$ n = 128 $ $ \varepsilon= 2^{-20} $ -
Figure 9.
Loglog plot of errors of the Problem 6.2 for
$ n = 64, 128, 256 $ $ \varepsilon= 2^{-10} $ -
Figure 10.
Loglog plot of errors of the Problem 6.2 for
$ n = 64, 128, 256 $ $ \varepsilon=2^{-20} $ -
Figure 11.
Approximate solutions of the Problem 6.2 at various time level with
$ n = 64 $ $ \varepsilon= 2^{-20} $ -
Figure 12.
Approximate solutions of the Problem 6.2 at various time level with
$ n =128 $ $ \varepsilon= 2^{-20} $