| Citation: |
Parvin Kumari, Devendra Kumar, Higinio Ramos. PARAMETER INDEPENDENT SCHEME FOR SINGULARLY PERTURBED PROBLEMS INCLUDING A BOUNDARY TURNING POINT OF MULTIPLICITY ≥ 1[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1304-1320. doi: 10.11948/20220123
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PARAMETER INDEPENDENT SCHEME FOR SINGULARLY PERTURBED PROBLEMS INCLUDING A BOUNDARY TURNING POINT OF MULTIPLICITY ≥ 1
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Abstract
A numerical scheme is developed for parabolic singularly perturbed boundary value problems, including multiple boundary turning points at the left endpoint of the spatial direction. The highest order derivative of these problems is multiplied by a small parameter $ \varepsilon\, (0< \varepsilon\ll 1) $, and when it is close to zero, the solution exhibits a parabolic type boundary layer near the left lateral surface of the domain of consideration. Thus, large oscillations appear when classical/standard numerical methods are used to solve the problem, and one cannot achieve the expected accuracy. Thus, the Crank-Nicolson scheme on a uniform mesh in the temporal direction and an upwind scheme on a Shishkin-type mesh in the spatial direction is constructed. The theoretical analysis shows that the method converges irrespective of the size of $ \varepsilon $ with accuracy $ \mathcal{O}((\Delta t)^2+N^{-1}\ln N) $. Three test examples are presented to verify that the computational results agree with the theoretical ones.
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References
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Parvin Kumari, Devendra Kumar, Higinio Ramos. PARAMETER INDEPENDENT SCHEME FOR SINGULARLY PERTURBED PROBLEMS INCLUDING A BOUNDARY TURNING POINT OF MULTIPLICITY ≥ 1[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1304-1320. doi: 10.11948/20220123
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Figure 1.
Numerical solution profiles for Example 5.1 for (a)
$ \varepsilon=1, p = 1 $ $ \varepsilon=2^{-6}, p = 3 $ $ \varepsilon=2^{-12}, p = 5 $ $ \varepsilon=2^{-18}, p = 7 $ -
Figure 2.
Numerical solution profiles for
$ p=2 $ $ \varepsilon=1 $ $ \varepsilon=0.1 $ $ \varepsilon=0.01 $ $ \varepsilon=0.001 $ -
Figure 3.
Numerical solution profiles for Example 5.2 for (a)
$ \varepsilon=1, p = 1 $ $ \varepsilon=2^{-6}, p = 3 $ $ \varepsilon=2^{-12}, p = 5 $ $ \varepsilon=2^{-18}, p = 7 $ -
Figure 4.
Numerical solution profiles for
$ p=2 $ $ \varepsilon=1 $ $ \varepsilon=0.1 $ $ \varepsilon=0.01 $ $ \varepsilon=0.001 $ -
Figure 5.
Surface plots of the numerical solution for Example 5.3 for
$ \varepsilon=2^{-8}, p = 1 $