# American Institute of Mathematical Sciences

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

 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.

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.
The Bay of Bengal and the coastal region of Bangladesh
Model domain
Color contours of $\eta_h^k$ by finite difference scheme (25) for the five cases $(i)$-$(v)$ discussed in Subsection 4.2
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 $(i)$-$(v)$
Color contours of $\eta_h^k$ by Lagrange-Galerkin scheme (26) for the five cases $(i)$-$(v)$ discussed in Section 5
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 $-6.75 \times 10^{-3}$ $-12.54$ $-5.01 \times 10^{-9}$ $-1.14 \times 10^{-7}$
 $\varGamma_T$ $\varGamma_D$ $I_{h1}$ $I_{h2}$ $I_{h3}$ $I_{h4}$ 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 $-6.75 \times 10^{-3}$ $-12.54$ $-5.01 \times 10^{-9}$ $-1.14 \times 10^{-7}$
$c_0$ 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
 $c_0$ $\mathcal{S}_h(c_0)$ 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
 [1] Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 [2] Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1 [3] Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317 [4] Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 [5] François Bouchut, Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 739-758. doi: 10.3934/dcdsb.2010.13.739 [6] Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835 [7] Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks and Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145 [8] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [9] Navnit Jha. Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization. Conference Publications, 2013, 2013 (special) : 355-363. doi: 10.3934/proc.2013.2013.355 [10] Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 79-93. doi: 10.3934/dcdss.2021030 [11] Hung Le. Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3357-3385. doi: 10.3934/dcds.2018144 [12] Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015 [13] Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593 [14] Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799 [15] Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, 2019, 6 (1) : 1-37. doi: 10.3934/jcd.2019001 [16] Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375 [17] Xiaoping Zhai, Hailong Ye. On global large energy solutions to the viscous shallow water equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4277-4293. doi: 10.3934/dcdsb.2020097 [18] Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327 [19] David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629 [20] Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331

2021 Impact Factor: 1.865