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March  2021, 14(3): 1063-1078. doi: 10.3934/dcdss.2020230

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

Md. Masum Murshed 1,2,, , Kouta Futai 1, , Masato Kimura 1, and  Hirofumi Notsu 1,3,

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

Received  January 2019 Revised  June 2019 Published  March 2021 Early access  December 2019

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.

Keywords: Shallow water equations, stability, transmission boundary condition, finite difference method, Lagrange-Galerkin scheme.
Mathematics Subject Classification: Primary: 65M06, 76D05, 65L07; Secondary: 65M25, 65M60.
Citation: Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230
References:
[1]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, Journal de Mathématiques Pures et Appliquées, 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.

[2]

P. K. Das, Prediction model for storm surges in the Bay of Bengal, Nature, 239 (1972), 211-213.  doi: 10.1038/239211a0.

[3]

S. K. Debsarma, Simulations of storm surges in the Bay of Bengal, Marine Geodesy, 32 (2009), 178-198.  doi: 10.1080/01490410902869458.

[4]

B. Jonhs and A. Ali, The numerical modeling of storm surges in the Bay of Bengal, Quarterly Journal of the Royal Meteorological Society, 106 (1980), 1-18. 

[5]

H. Kanayama and H. Dan, A finite element scheme for two-layer viscous shallow-water equations, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 163-191.  doi: 10.1007/BF03167549.

[6]

H. Kanayama and H. Dan, Tsunami propagation from the open sea to the coast, Tsunami, Chapter 4, IntechOpen, (2016), 61-72.

[7]

H. Kanayama and T. Ushijima, On the viscous shallow-water equations. Ⅰ. Derivation and conservation laws, Memoirs of Numerical Mathematics, (1981/82), 39-64.

[8]

H. Kanayama and T. Ushijima, On the viscous shallow-water equations. Ⅱ. A linearized system, Bulletin of University of Electro-Communications, 1 (1988), 347-355. 

[9]

H. Kanayama and T. Ushijima, On the viscous shallow-water equations. Ⅲ. A finite element scheme, Bulletin of University of Electro-Communications, 2 (1989), 47-62. 

[10]

C. Lucas, Cosine effect on shallow water equations and mathematical properties, Quarterly of Applied Mathematics, American Mathematical Society, 67 (2009), 283-310.  doi: 10.1090/S0033-569X-09-01113-0.

[11]

G. C. Paul and A. I. M. Ismail, Tide surge interaction model including air bubble effects for the coast of Bangladesh, Journal of the Franklin Institute, 349 (2012), 2530-2546.  doi: 10.1016/j.jfranklin.2012.08.003.

[12]

G. C. Paul and A. I. M. Ismail, Contribution of offshore islands in the prediction of water levels due to tide-surge interaction for the coastal region of Bangladesh, Natural Hazards, 65 (2013), 13-25.  doi: 10.1007/s11069-012-0341-z.

[13]

G. C. Paul, A. I. M. Ismail and M. F. Karim, Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh, Journal of Oceanography, 70 (2014), 199-210.  doi: 10.1007/s10872-014-0224-x.

[14]

G. C. Paul, M. M. Murshed, M. R. Haque, M. M. Rahman and A. Hoque, Development of a cylindrical polar coordinates shallow water storm surge model for the coast of Bangladesh, Journal of Coastal Conservation, 21 (2017), 951-966.  doi: 10.1007/s11852-017-0565-x.

[15]

G. C. Paul, S. Senthilkumar and R. Pria, Storm surge simulation along the Meghna estuarine area: An alternative approach, Acta Oceanologica Sinica, 37 (2018), 40-49.  doi: 10.1007/s13131-018-1157-9.

[16]

G. C. Paul, S Senthilkumar and R. Pria, An efficient approach to forecast water levels owing to the interaction of tide and surge associated with a storm along the coast of Bangladesh, Ocean Engineering, 148 (2018), 516-529.  doi: 10.1016/j.oceaneng.2017.10.031.

[17]

G. D. Roy, A. B. M. Humayun Kabir, M. M. Mandal and M. Z. Haque, Polar coordinate shallow water storm surge model for the coast of Bangladesh, Dynamics of Atmospheres and Oceans, 29 (1999), 397-413.  doi: 10.1016/S0377-0265(99)00012-3.

[18]

G. D. Roy and A. B. H. M. Kabir, Use of nested numerical scheme in a shallow water model for the coast of Bangladesh, BRAC University Journal, 1 (2004), 79-92. 

[19]

H. X. Rui and M. Tabata, A mass-conservative characteristic finite element scheme for convection-diffusion problems, Journal of Scientific Computing, 43 (2010), 416-432.  doi: 10.1007/s10915-009-9283-3.

show all references

References:
[1]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, Journal de Mathématiques Pures et Appliquées, 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.

[2]

P. K. Das, Prediction model for storm surges in the Bay of Bengal, Nature, 239 (1972), 211-213.  doi: 10.1038/239211a0.

[3]

S. K. Debsarma, Simulations of storm surges in the Bay of Bengal, Marine Geodesy, 32 (2009), 178-198.  doi: 10.1080/01490410902869458.

[4]

B. Jonhs and A. Ali, The numerical modeling of storm surges in the Bay of Bengal, Quarterly Journal of the Royal Meteorological Society, 106 (1980), 1-18. 

[5]

H. Kanayama and H. Dan, A finite element scheme for two-layer viscous shallow-water equations, Japan Journal of Industrial and Applied Mathematics, 23 (2006), 163-191.  doi: 10.1007/BF03167549.

[6]

H. Kanayama and H. Dan, Tsunami propagation from the open sea to the coast, Tsunami, Chapter 4, IntechOpen, (2016), 61-72.

[7]

H. Kanayama and T. Ushijima, On the viscous shallow-water equations. Ⅰ. Derivation and conservation laws, Memoirs of Numerical Mathematics, (1981/82), 39-64.

[8]

H. Kanayama and T. Ushijima, On the viscous shallow-water equations. Ⅱ. A linearized system, Bulletin of University of Electro-Communications, 1 (1988), 347-355. 

[9]

H. Kanayama and T. Ushijima, On the viscous shallow-water equations. Ⅲ. A finite element scheme, Bulletin of University of Electro-Communications, 2 (1989), 47-62. 

[10]

C. Lucas, Cosine effect on shallow water equations and mathematical properties, Quarterly of Applied Mathematics, American Mathematical Society, 67 (2009), 283-310.  doi: 10.1090/S0033-569X-09-01113-0.

[11]

G. C. Paul and A. I. M. Ismail, Tide surge interaction model including air bubble effects for the coast of Bangladesh, Journal of the Franklin Institute, 349 (2012), 2530-2546.  doi: 10.1016/j.jfranklin.2012.08.003.

[12]

G. C. Paul and A. I. M. Ismail, Contribution of offshore islands in the prediction of water levels due to tide-surge interaction for the coastal region of Bangladesh, Natural Hazards, 65 (2013), 13-25.  doi: 10.1007/s11069-012-0341-z.

[13]

G. C. Paul, A. I. M. Ismail and M. F. Karim, Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh, Journal of Oceanography, 70 (2014), 199-210.  doi: 10.1007/s10872-014-0224-x.

[14]

G. C. Paul, M. M. Murshed, M. R. Haque, M. M. Rahman and A. Hoque, Development of a cylindrical polar coordinates shallow water storm surge model for the coast of Bangladesh, Journal of Coastal Conservation, 21 (2017), 951-966.  doi: 10.1007/s11852-017-0565-x.

[15]

G. C. Paul, S. Senthilkumar and R. Pria, Storm surge simulation along the Meghna estuarine area: An alternative approach, Acta Oceanologica Sinica, 37 (2018), 40-49.  doi: 10.1007/s13131-018-1157-9.

[16]

G. C. Paul, S Senthilkumar and R. Pria, An efficient approach to forecast water levels owing to the interaction of tide and surge associated with a storm along the coast of Bangladesh, Ocean Engineering, 148 (2018), 516-529.  doi: 10.1016/j.oceaneng.2017.10.031.

[17]

G. D. Roy, A. B. M. Humayun Kabir, M. M. Mandal and M. Z. Haque, Polar coordinate shallow water storm surge model for the coast of Bangladesh, Dynamics of Atmospheres and Oceans, 29 (1999), 397-413.  doi: 10.1016/S0377-0265(99)00012-3.

[18]

G. D. Roy and A. B. H. M. Kabir, Use of nested numerical scheme in a shallow water model for the coast of Bangladesh, BRAC University Journal, 1 (2004), 79-92. 

[19]

H. X. Rui and M. Tabata, A mass-conservative characteristic finite element scheme for convection-diffusion problems, Journal of Scientific Computing, 43 (2010), 416-432.  doi: 10.1007/s10915-009-9283-3.

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
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
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
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
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
Min $ -6.75 \times 10^{-3} $ $ -12.54 $ $ -5.01 \times 10^{-9} $ $ -1.14 \times 10^{-7} $
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$ \varGamma_T $ $ \varGamma_D $ $ I_{h1} $ $ I_{h2} $ $ I_{h3} $ $ I_{h4} $
0 4 Max 0.00 0.00 0.00 0.00
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
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
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
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
Min $ -6.75 \times 10^{-3} $ $ -12.54 $ $ -5.01 \times 10^{-9} $ $ -1.14 \times 10^{-7} $
Table 2.  $ c_0 $ and $ \mathcal{S}_h(c_0) $
$ c_0 $ $ \mathcal{S}_h(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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$ c_0 $ $ \mathcal{S}_h(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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