Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition

Md. Masum Murshed; Kouta Futai; Masato Kimura; Hirofumi Notsu

doi:10.3934/dcdss.2020230

Article Contents

2021, Volume 14, Issue 3: 1063-1078. Doi: 10.3934/dcdss.2020230

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Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition

1.
Division of Mathematical and Physical Sciences, Kanazawa University, Kanazawa 920-1192, Japan
2.
Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh
3.
Japan Science and Technology Agency, PRESTO, Kawaguchi 332-0012, Japan

^* Corresponding author: Md. Masum Murshed
^* Corresponding author: Md. Masum Murshed

Received: January 2019

Revised: June 2019

Early access: December 2019

Published: March 2021

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Abstract

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirichlet and the slip boundary conditions. For the SWEs with a transmission boundary condition, an inequality for the energy estimate is proved under some assumptions to be satisfied in practical computation. In the numerical part, based on the theoretical results, the energy estimate of the SWEs with a transmission boundary condition is confirmed numerically by a finite difference method (FDM). The choice of a positive constant $ c_0 $ used in the transmission boundary condition is investigated additionally. Furthermore, we present numerical results by a Lagrange-Galerkin scheme, which are similar to those by the FDM. The theoretical results along with the numerical results strongly recommend that the transmission boundary condition is suitable for the boundaries in the open sea.

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Mathematics Subject Classification: Primary: 65M06, 76D05, 65L07; Secondary: 65M25, 65M60.

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Figure 1. The Bay of Bengal and the coastal region of Bangladesh

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Figure 2. Model domain

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Figure 3. Color contours of $ \eta_h^k $ by finite difference scheme (25) for the five cases $ (i) $-$ (v) $ discussed in Subsection 4.2

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Figure 4. Graphs of $ E_h^k $ (left), $ \sum_{i = 1}^4 I_{hi}^k \approx \frac{d}{dt} E(t) $ (center) and $ I_{hi}^k $, $ i = 1, \ldots, 4 $, (right) versus $ t = t^k\; (\ge 0, k\in \mathbb{Z}) $ for the five cases $ (i) $-$ (v) $

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Figure 5. Color contours of $ \eta_h^k $ by Lagrange-Galerkin scheme (26) for the five cases $ (i) $-$ (v) $ discussed in Section 5

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Table 1. Maximum and minimum values of $ I_{hi}^k $, $ i = 1,\ldots, 4 $, with respect to the number of transmission boundaries

$ \varGamma_T $	$ \varGamma_D $		$ I_{h1} $	$ I_{h2} $	$ I_{h3} $	$ I_{h4} $
0	4	Max	0.00	0.00	0.00	0.00
0	4	Min	0.00	0.00	0.00	$ -8.63 \times 10^{-7} $
1	3	Max	$ 1.10 \times 10^{-4} $	0.00	$ 1.44 \times 10^{-9} $	0.00
1	3	Min	$ -2.59 \times 10^{-3} $	$ -3.37 $	$ -1.25\times 10^{-9} $	$ -3.76\times 10^{-7} $
2	2	Max	$ 1.86 \times 10^{-4} $	0.00	$ 1.72 \times 10^{-9} $	0.00
2	2	Min	$ -3.38 \times 10^{-3} $	-6.27	$ -2.50 \times 10^{-9} $	$ -2.31 \times 10^{-7} $
3	1	Max	$ 1.43 \times 10^{-4} $	0.00	$ 2.58 \times 10^{-9} $	0.00
3	1	Min	$ -5.06 \times 10^{-3} $	$ -9.40 $	$ -3.75 \times 10^{-9} $	$ -1.74 \times 10^{-7} $
4	0	Max	$ 2.87 \times 10^{-4} $	0.00	$ 3.47 \times 10^{-9} $	0.00
4	0	Min	$ -6.75 \times 10^{-3} $	$ -12.54 $	$ -5.01 \times 10^{-9} $	$ -1.14 \times 10^{-7} $

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Table 2. $ c_0 $ and $ \mathcal{S}_h(c_0) $

$ c_0 $	$ \mathcal{S}_h(c_0) $
$ c_0 $	Case Ⅰ	Case Ⅱ	Case Ⅲ	Case Ⅳ	Case Ⅴ	Case Ⅵ
0.1	12.17	8.53	8.14	5.47	44.48	44.49
0.2	9.89	6.97	6.35	4.04	34.36	34.37
0.3	8.84	6.24	5.52	3.35	28.74	28.75
0.4	8.27	5.85	5.08	2.98	25.23	25.24
0.5	7.93	5.61	4.84	2.79	22.82	22.83
0.6	7.71	5.46	4.71	2.69	21.05	21.06
0.7	7.58	5.37	4.65	2.66	19.69	19.69
0.8	7.51	5.32	4.63	2.67	18.60	18.61
0.9	7.4805	5.2969	4.64	2.70	17.71	17.72
1.0	7.4807	5.2977	4.68	2.76	16.98	16.98
1.1	7.50	5.32	4.73	2.82	16.36	16.36
1.2	7.55	5.35	4.79	2.89	15.83	15.84
1.5	7.75	5.49	5.02	3.12	14.66	14.66

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