# A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography

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## Abstract

**:**

## 1. Introduction

## 2. A Two-Step Lagrange–Galerkin Scheme

#### 2.1. Statement of the Problem

#### 2.2. Presentation of the Scheme

**Remark**

**1.**

**Remark**

**2.**

- $\left(i\right)$
- At each time step, we obtain ${\varphi}_{h}^{n}\in {\Psi}_{h}$ from Equation (5a) and ${u}_{h}^{n}\in {V}_{h}\left(G({\varphi}_{h}^{n};{\eta}_{h}^{n})\right)$ from Equation (5b) combined with Equation(5c), where both of the resulting coefficient matrices of the systems of linear equations derived from Equations (5a) and (5b) are symmetric.
- $\left(ii\right)$
- We need ${\mathcal{A}}_{\Delta t}^{\left(1\right)}\left[u\right]$ and ${\mathcal{B}}_{\Delta t}^{\left(1\right)}\left[w\right]$ due to the lack of the functions ${\varphi}_{h}^{n-2}$ and ${u}_{h}^{n-2}$ for $n=1$, which are used for ${\mathcal{A}}_{\Delta t}^{\left(2\right)}\left[{u}_{h}\right]{\varphi}_{h}^{n}$ and ${\mathcal{B}}_{\Delta t}^{\left(2\right)}\left[{u}_{h}\right]{u}_{h}^{n}$ for $n\ge 2$.
- $\left(iii\right)$
- $\left(iv\right)$
- It is discussed in [40,49] that the one-time use of first-order single-step methods, ${\mathcal{A}}_{\Delta t}^{\left(1\right)}\left[{u}_{h}\right]{\varphi}_{h}^{n}$ and ${\mathcal{B}}_{\Delta t}^{\left(1\right)}\left[{u}_{h}\right]{u}_{h}^{n}$, has no loss of convergence order in discrete version of ${L}^{\infty}(0,T;{L}^{2}(\Omega ))$-norm for a conservative convection-diffusion equation and the Navier–Stokes equations, respectively.
- $\left(v\right)$
- The so-called quadrilateral elements ${Q}_{k}\left(K\right)$, e.g., bilinear $(k=1)$ and biquadratic $(k=2)$ elements, with a partition of $\overline{\Omega}$, ${\mathcal{T}}_{h}=\left\{K\right\}$, by rectangles are also available for ${\Psi}_{h}$ and ${Y}_{h}$.

**Remark**

**3.**

**Remark**

**4.**

## 3. Numerical Results in Square Domains

#### 3.1. Experimental Order of Convergence

**Example**

**1**

**.**In problem (3), we set $\Omega ={(0,1)}^{2}$, $\Gamma ={\Gamma}_{\mathrm{D}}$ $({\Gamma}_{\mathrm{T}}=\varnothing )$, $T=1$, $g=\rho =\mu =\zeta =1$, and the function ${\eta}^{0}$, ${u}^{0}$, f and F are given so that the exact solution is

**Example**

**2**

**.**In Example 1, we replace ${\Gamma}_{\mathrm{T}}$ and ${\Gamma}_{\mathrm{D}}$ with ${\Gamma}_{\mathrm{T}}=\{x\in \Gamma ;\phantom{\rule{4pt}{0ex}}{x}_{2}=0\}$ and ${\Gamma}_{\mathrm{D}}=\Gamma \backslash {\overline{\Gamma}}_{\mathrm{T}}$, respectively.

**Remark**

**5.**

#### 3.2. Effect of the TBC

**Example**

**3.**

- (a)
- ${\Gamma}_{\mathrm{T}}=\varnothing $, i.e., $\Gamma ={\Gamma}_{\mathrm{D}}$,
- (b)
- ${\Gamma}_{\mathrm{T}}=\{x\in \Gamma ;\phantom{\rule{4pt}{0ex}}{x}_{2}=0\}$ (bottom), ${\Gamma}_{\mathrm{D}}=\Gamma \backslash {\overline{\Gamma}}_{\mathrm{T}}$,
- (c)
- ${\Gamma}_{\mathrm{T}}=\{x\in \Gamma ;\phantom{\rule{4pt}{0ex}}{x}_{1}=10,{x}_{2}=0\}$ (right and bottom), ${\Gamma}_{\mathrm{D}}=\Gamma \backslash {\overline{\Gamma}}_{\mathrm{T}}$,
- (d)
- ${\Gamma}_{\mathrm{T}}=\{x\in \Gamma ;\phantom{\rule{4pt}{0ex}}{x}_{1}=10,{x}_{2}=0,10\}$ (right, bottom and top), ${\Gamma}_{\mathrm{D}}=\Gamma \backslash {\overline{\Gamma}}_{\mathrm{T}}$,
- (e)
- ${\Gamma}_{\mathrm{T}}=\Gamma $.

## 4. Application to the Bay of Bengal

#### 4.1. Numerical Simulation with and without the TBC

**Example**

**4.**

**Remark**

**6.**

- (i)
- The last term in the RHS of (7) is obviously non-positive. From the TBC Equation (3e), i.e., $\varphi u={c}_{0}\sqrt{g\zeta}\eta n$, we observe that the second term in the RHS of Equation (7) is non-positive:$$-\rho g{\int}_{{\Gamma}_{\mathrm{T}}}\varphi \eta u\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}n\phantom{\rule{0.166667em}{0ex}}ds=-\rho g{\int}_{{\Gamma}_{\mathrm{T}}}{c}_{0}\sqrt{g\zeta}\phantom{\rule{0.166667em}{0ex}}{\eta}^{2}\phantom{\rule{0.166667em}{0ex}}ds\le 0.$$
- (ii)
- Let us additionally introduce a theorem ([9] (Theorem 3.4)). Suppose that there exists $\alpha \in (0,1)$ such that$$\begin{array}{cc}\hfill \eta (x,t)& \ge -\alpha \zeta \left(x\right)\phantom{\rule{1.em}{0ex}}(x\in {\overline{\Gamma}}_{\mathrm{T}},\phantom{\rule{4pt}{0ex}}t\in [0,T]),\hfill \\ \hfill 0& <{c}_{0}\le \sqrt{2/\alpha}\phantom{\rule{0.166667em}{0ex}}(1-\alpha ).\hfill \end{array}$$Then, the summation of the first and second terms in the RHS of Equation (7) is non-positive, i.e.,$$\begin{array}{c}\hfill -\frac{\rho}{2}{\int}_{{\Gamma}_{\mathrm{T}}}\varphi {\left|u\right|}^{2}u\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}n\phantom{\rule{0.166667em}{0ex}}ds-\rho g{\int}_{{\Gamma}_{\mathrm{T}}}\varphi \eta u\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}n\phantom{\rule{0.166667em}{0ex}}ds\le 0,\end{array}$$$$\begin{array}{cc}\hfill \frac{d}{dt}\mathcal{E}\left(t\right)& \le 2\mu {\int}_{{\Gamma}_{\mathrm{T}}}\varphi \left[D\left(u\right)n\right]\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.166667em}{0ex}}ds.\hfill \end{array}$$
- (iii)
- (iv)
- We have compared our results by the TBC (${c}_{0}=0.9$) and a modified TBC (${c}_{0}=1$) with those by the RBC:$$\begin{array}{c}\hfill u=\sqrt{g/\zeta}\phantom{\rule{0.166667em}{0ex}}\eta n\phantom{\rule{2.em}{0ex}}on\phantom{\rule{3.33333pt}{0ex}}{\Gamma}_{\mathrm{T}}\times (0,T).\end{array}$$The results by the three boundary conditions are not significantly different as presented in Appendix B. We note that condition (9) is the well-known RBC, cf., e.g., [8], and that the modified TBC is obtained by replacing ζ with ${\varphi}^{2}/\zeta $ in Equation (9), where the relation $\zeta \approx {\varphi}^{2}/\zeta \approx \varphi $ holds if $\left|\eta \right|\ll \zeta $ is satisfied.

#### 4.2. Effect of Position of a Transmission Boundary

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RBC | Radiation boundary condition |

TBC | Transmission boundary condition |

SWEs | Shallow water equations |

EOC | Experimental order of convergence |

LG1 | Single-step Lagrange–Galerkin scheme of first order in time |

LG2 | Two-step Lagrange–Galerkin scheme of second order in time |

## Appendix A. Choice of c_{0}

- -
- Case I (the square domain). In problem (3), we set $\Omega ={(0,10)}^{2}$, $T=100$, $g=9.8\times {10}^{-3}$, $\rho ={10}^{12}$, $\mu =\zeta =1$, $(f,F)=(0,0)$, $c={10}^{-3}$, ${\eta}^{0}={cexp(-100|x-p|}^{2})$, $p={(5,5)}^{\top}$, ${u}^{0}=0$ and $\Gamma ={\Gamma}_{\mathrm{T}}$ $({\Gamma}_{\mathrm{D}}=\varnothing )$. We employ discretization parameters, $N=200$ $(h=1/N)$, and $\Delta t=0.25\sqrt{h}$.
- -
- Case II (the Bay of Bengal). The parameters are the same as Example 4 except the value of ${c}_{0}$. We employ the same mesh and $\Delta t\phantom{\rule{3.33333pt}{0ex}}(=0.2)$ in Section 4.

**Table A1.**Values of ${c}_{0}$ and $\parallel {\eta}_{h}{\parallel}_{{\ell}^{2}\left({L}^{2}\right)}$.

$\parallel {\mathit{\eta}}_{\mathit{h}}{\parallel}_{{\mathit{\ell}}^{2}\left({\mathit{L}}^{2}\right)}$ | ||
---|---|---|

Value of ${\mathit{c}}_{0}$ | Case I (the Square Domain) | Case II (the Bay of Bengal) |

$0.5$ | $8.16\phantom{0}\times {10}^{-2}$ | 13.55 |

$0.6$ | $8.08\phantom{0}\times {10}^{-2}$ | 13.54 |

$0.7$ | $8.03\phantom{0}\times {10}^{-2}$ | 13.5342 |

$0.8$ | $8.002\times {10}^{-2}$ | 13.5323 |

$0.9$ | $7.997\times {10}^{-2}$ | 13.5319 |

$1.0$ | $8.006\times {10}^{-2}$ | 13.5328 |

$1.1$ | $8.02\phantom{4}\times {10}^{-2}$ | 13.5354 |

$1.2$ | $8.05\phantom{0}\times {10}^{-2}$ | 13.5375 |

## Appendix B. Comparison with Radiation Type Open Boundary Condition

**Table A2.**Valuesof $\parallel {\eta}_{h}{\parallel}_{{\ell}^{2}\left({L}^{2}\right)}$ for different boundary conditions for Case I and Case II.

$\parallel {\mathit{\eta}}_{\mathit{h}}{\parallel}_{{\mathit{\ell}}^{2}\left({\mathit{L}}^{2}\right)}$ | ||
---|---|---|

Boundary Condition | Case I (the Square Domain) | Case II (the Bay of Bengal) |

TBC | $7.997\times {10}^{-2}$ | 13.5316 |

modified TBC with ${c}_{0}=1$ | $8.006\times {10}^{-2}$ | 13.5328 |

RBC | $8.007\times {10}^{-2}$ | 13.5334 |

**Figure A1.**Graphs of ${L}^{2}(\Omega )$-norm of ${\eta}_{h}^{n}$ for different boundary conditions for Case II (the Bay of Bengal).

## References

- Debsarma, S.K. Simulations of storm surges in the Bay of Bengal. Mar. Geod.
**2009**, 32, 178–198. [Google Scholar] [CrossRef] - Das, P.K. Prediction Model for Storm Surges in the Bay of Bengal. Nature
**1972**, 239, 211–213. [Google Scholar] [CrossRef] - Johns, B. Numerical simulation of storm surges in the Bay of Bengal. In Monsoon Dynamics; Cambridge University Press: Cambridge, UK, 1981; pp. 689–706. [Google Scholar]
- Roy, G.; Kabir, A.H.; Mandal, M.; Haque, M. Polar coordinates shallow water storm surge model for the coast of Bangladesh. Dyn. Atmos. Ocean.
**1999**, 29, 397–413. [Google Scholar] [CrossRef] - Paul, G.C.; Ismail, A.I.M. Tide–surge interaction model including air bubble effects for the coast of Bangladesh. J. Frankl. Inst.
**2012**, 349, 2530–2546. [Google Scholar] [CrossRef] - Paul, G.C.; Ismail, A.I.M. Contribution of offshore islands in the prediction of water levels due to tide–surge interaction for the coastal region of Bangladesh. Nat. Hazards
**2013**, 65, 13–25. [Google Scholar] [CrossRef] - Paul, G.C.; Senthilkumar, S.; Pria, R. Storm surge simulation along the Meghna estuarine area: An alternative approach. Acta Oceanol. Sin.
**2018**, 37, 40–49. [Google Scholar] [CrossRef] - Dube, S.; Sinha, P.; Roy, G. Numerical simulation of storm surges in Bangladesh using a bay-river coupled model. Coast. Eng.
**1986**, 10, 85–101. [Google Scholar] [CrossRef] - Murshed, M.M.; Futai, K.; Kimura, M.; Notsu, H. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discret. Contin. Dyn. Syst.-S
**2021**, 14, 1063–1078. [Google Scholar] [CrossRef][Green Version] - Murshed, M.M. Theoretical and Numerical Studies of the Shallow Water Equations with a Transmission Boundary Condition. Ph.D. Thesis, Kanazawa University, Kanazawa, Japan, 2019. [Google Scholar]
- Sommerfeld, A. Partial Differential Equations: Lectures in Theoretical Physics; Academic Press: Cambridge, MA, USA, 1949; Volume 6. [Google Scholar]
- Orlanski, I. A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys.
**1976**, 21, 251–269. [Google Scholar] [CrossRef] - Røed, L.P.; Cooper, C.K. Open Boundary Conditions in Numerical Ocean Models. Adv. Phys. Oceanogr. Numer. Model.
**1986**, 186, 411–436. [Google Scholar] - Jensen, T.G. Open boundary conditions in stratified ocean models. J. Mar. Syst.
**1998**, 16, 297–322. [Google Scholar] [CrossRef] - Kanayama, H.; Dan, H. Tsunami Propagation from the open sea to the coast. In Tsunami; IntechOpen: London, UK, 2016. [Google Scholar]
- Kanayama, H.; Dan, H. A finite element scheme for two-layer viscous shallow-water equations. Jpn. J. Ind. Appl. Math.
**2006**, 23, 163–191. [Google Scholar] [CrossRef] - Ewing, R.; Russell, T. Multistep Galerkin methods along characteristics for convection-diffusion problems. In Advances in Computer Methods for Partial Differential Equations IV; Vichnevetsky, R., Stepleman, R., Eds.; IMACS: New Brunswick, NJ, USA, 1981; pp. 28–36. [Google Scholar]
- Douglas, J.J.; Russell, T.F. Numerical Methods for Convection-Dominated Diffusion Problems Based on Combining the Method of Characteristics with Finite Element or Finite Difference Procedures. SIAM J. Numer. Anal.
**1982**, 19, 871–885. [Google Scholar] [CrossRef] - Pironneau, O. On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math.
**1982**, 38, 309–332. [Google Scholar] [CrossRef] - Rui, H.; Tabata, M. A mass-conservative characteristic finite element scheme for convection-diffusion problems. J. Sci. Comput.
**2010**, 43, 416–432. [Google Scholar] [CrossRef][Green Version] - Ewing, R.; Russell, T.; Wheeler, M. Simulation of miscible displacement using mixed methods and a modified method of characteristics. In Proceedings of the Seventh Reservoir Simulation Symposium; Society of Petroleum Engineers of AIME: San Francisco, CA, USA, 1983; pp. 71–81. [Google Scholar]
- Süli, E. Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math.
**1988**, 53, 459–483. [Google Scholar] [CrossRef] - Pironneau, O. Finite Element Methods for Fluids; John Wiley & Sons: Chichester, UK, 1989. [Google Scholar]
- Boukir, K.; Maday, Y.; Métivet, B.; Razafindrakoto, E. A high-order characteristics/finite element method for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids
**1997**, 25, 1421–1454. [Google Scholar] [CrossRef] - Achdou, Y.; Guermond, J.L. Convergence analysis of a finite element projection/Lagrange–Galerkin method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal.
**2000**, 37, 799–826. [Google Scholar] [CrossRef] - Rui, H.; Tabata, M. A second order characteristic finite element scheme for convection-diffusion problems. Numer. Math.
**2002**, 92, 161–177. [Google Scholar] [CrossRef] - Bermúdez, A.; Nogueiras, M.R.; Vázquez, C. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part I: Time Discretization. SIAM J. Numer. Anal.
**2006**, 44, 1829–1853. [Google Scholar] [CrossRef] - Bermúdez, A.; Nogueiras, M.R.; Vázquez, C. Numerical analysis of convection-diffusion-reaction problems with higher order characteristics/finite elements. Part II: Fully discretized scheme and quadrature formulas. SIAM J. Numer. Anal.
**2006**, 44, 1854–1876. [Google Scholar] [CrossRef] - Chrysafinos, K.; Walkington, N.J. Lagrangian and moving mesh methods for the convection diffusion equation. ESAIM Math. Model. Numer. Anal.
**2008**, 42, 25–55. [Google Scholar] [CrossRef][Green Version] - Notsu, H. Numerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme. Trans. Jpn. Soc. Comput. Eng. Sci.
**2008**, 2008, 20080032. [Google Scholar] - Pironneau, O.; Tabata, M. Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type. Int. J. Numer. Methods Fluids
**2010**, 64, 1240–1253. [Google Scholar] [CrossRef] - Benítez, M.; Bermúdez, A. A second order characteristics finite element scheme for natural convection problems. J. Comput. Appl. Math.
**2011**, 235, 3270–3284. [Google Scholar] [CrossRef][Green Version] - Benítez, M.; Bermúdez, A. Numerical analysis of a second order pure Lagrange–Galerkin method for convection-diffusion problems. Part I: Time discretization. SIAM J. Numer. Anal.
**2012**, 50, 858–882. [Google Scholar] [CrossRef][Green Version] - Benítez, M.; Bermúdez, A. Numerical analysis of a second order pure Lagrange–Galerkin method for convection-diffusion problems. Part II: Fully discretized scheme and numerical results. SIAM J. Numer. Anal.
**2012**, 50, 2824–2844. [Google Scholar] [CrossRef][Green Version] - Bermejo, R.; Saavedra, L. Modified Lagrange–Galerkin methods of first and second order in time for convection-diffusion problems. Numer. Mathematik
**2012**, 120, 601–638. [Google Scholar] [CrossRef] - Bermejo, R.; Gálan del Sastre, P.; Saavedra, L. A second order in time modified Lagrange–Galerkin finite element method for the incompressible Navier–Stokes equations. SIAM J. Numer. Anal.
**2012**, 50, 3084–3109. [Google Scholar] [CrossRef][Green Version] - Notsu, H.; Rui, H.; Tabata, M. Development and L2-Analysis of a Single-Step Characteristics Finite Difference Scheme of Second Order in Time for Convection-Diffusion Problems. J. Algorithms Comput. Technol.
**2013**, 7, 343–380. [Google Scholar] [CrossRef] - Notsu, H.; Tabata, M. Error Estimates of a Pressure-Stabilized Characteristics Finite Element Scheme for the Oseen Equations. J. Sci. Comput.
**2015**, 65, 940–955. [Google Scholar] [CrossRef] - Notsu, H.; Tabata, M. Error estimates of a stabilized Lagrange–Galerkin scheme for the Navier–Stokes equations. ESAIM Math. Model. Numer. Anal.
**2016**, 50, 361–380. [Google Scholar] [CrossRef][Green Version] - Notsu, H.; Tabata, M. Error Estimates of a Stabilized Lagrange–Galerkin Scheme of Second-Order in Time for the Navier–Stokes Equations. In Mathematical Fluid Dynamics, Present and Future Springer Proceedings in Mathematics & Statistics; Springer: Berlin, Germany, 2016; pp. 497–530. [Google Scholar]
- Tabata, M.; Uchiumi, S. A genuinely stable Lagrange–Galerkin scheme for convection-diffusion problems. Jpn. J. Ind. Appl. Math.
**2016**, 33, 121–143. [Google Scholar] [CrossRef][Green Version] - Lukáčová-Medviďová, M.; Notsu, H.; She, B. Energy dissipative characteristic schemes for the diffusive Oldroyd-B viscoelastic fluid. Int. J. Numer. Methods Fluids
**2015**, 81, 523–557. [Google Scholar] [CrossRef] - Lukáčová-Medvid’ová, M.; Mizerová, H.; Notsu, H.; Tabata, M. Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method, Part I: A linear scheme. ESAIM Math. Model. Numer. Anal.
**2017**, 51, 1637–1661. [Google Scholar] [CrossRef][Green Version] - Lukáčová-Medvid’ová, M.; Mizerová, H.; Notsu, H.; Tabata, M. Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange–Galerkin method, Part II: A nonlinear scheme. ESAIM Math. Model. Numer. Anal.
**2017**, 51, 1663–1689. [Google Scholar] [CrossRef][Green Version] - Tabata, M.; Uchiumi, S. An exactly computable Lagrange–Galerkin scheme for the Navier–Stokes equations and its error estimates. Math. Comput.
**2018**, 87, 39–67. [Google Scholar] [CrossRef] - Uchiumi, S. A viscosity-independent error estimate of a pressure-stabilized Lagrange–Galerkin scheme for the Oseen problem. J. Sci. Comput.
**2019**, 80, 834–858. [Google Scholar] [CrossRef][Green Version] - Colera, M.; Carpio, J.; Bermejo, R. A nearly-conservative high-order Lagrange–Galerkin method for the resolution of scalar convection-dominated equations in non-divergence-free velocity fields. Comput. Methods Appl. Mech. Eng.
**2020**, 372, 113366. [Google Scholar] [CrossRef] - Colera, M.; Carpio, J.; Bermejo, R. A nearly-conservative, high-order, forward Lagrange–Galerkin method for the resolution of scalar hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng.
**2021**, 376, 113654. [Google Scholar] [CrossRef] - Futai, K.; Kolbe, N.; Notsu, H.; Suzuki, T. A mass-preserving two-step Lagrange–Galerkin scheme for convection-diffusion problems. J. Sci. Comput.
**2022**, 92, 37. [Google Scholar] [CrossRef] - Hecht, F. New development in FreeFem++. J. Numer. Math.
**2012**, 20, 251–265. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Diagrams for the problem; left: the domain $\Omega $ and the velocity $u={({u}_{1},{u}_{2})}^{\top}$; right: the total wave height $\varphi =\eta +\zeta $.

**Figure 4.**Graphs of errors ${E}_{0}\left(\eta \right)$ and ${E}_{0}\left(u\right)$ in logarithmic scale by LG1 for Example 1 (

**i**) and Example 2 (

**ii**), and by LG2 for Example 1 (

**iii**) and Example 2 (

**iv**).

**Figure 5.**Graphs of errors ${E}_{1}\left(\eta \right)$ and ${E}_{1}\left(u\right)$ in logarithmic scale by LG1 for Example 1 (

**i**) and Example 2 (

**ii**), and by LG2 for Example 1 (

**iii**) and Example 2 (

**iv**).

**Figure 6.**Color contours of ${\eta}_{h}^{n}$ by LG2 with and without the TBC for the five cases, (

**a**–

**e**), in Example 3.

**Figure 7.**The domain for the Bay of Bengal region with the information of boundaries, ${\Gamma}_{\mathrm{D}}$ and ${\Gamma}_{\mathrm{T}}\phantom{\rule{3.33333pt}{0ex}}(={\Gamma}_{\mathrm{T}1}\cup {\Gamma}_{\mathrm{T}2}\cup {\Gamma}_{\mathrm{T}3})$ used in Example 4.

**Figure 9.**Contour plot of ${\eta}_{h}^{n}$ by LG2 with $\Gamma ={\Gamma}_{\mathrm{D}}$ (

**left**) and $\Gamma ={\Gamma}_{\mathrm{D}}\cup {\Gamma}_{\mathrm{T}}$ (

**right**) on the Bay of Bengal for $t=0,2500$ and $3000$.

**Figure 10.**Contour plot of ${\eta}_{h}^{n}$ by LG2 with $\Gamma ={\Gamma}_{\mathrm{D}}$ (

**left**) and $\Gamma ={\overline{\Gamma}}_{\mathrm{D}}\cup {\overline{\Gamma}}_{\mathrm{T}}$ (

**right**) on the Bay of Bengal for $t=4000,4500$ and $5000$.

**Figure 11.**Graphs of $\parallel {\eta}_{h}^{n}{\parallel}_{{L}^{2}(\Omega )}$ with respect to time $(t={t}^{n})$ for Example 4 with ${\Gamma}_{\mathrm{T}}$$(\Gamma ={\Gamma}_{\mathrm{D}}\cup {\Gamma}_{\mathrm{T}})$ and without ${\Gamma}_{\mathrm{T}}$$(\Gamma ={\Gamma}_{\mathrm{D}})$.

**Figure 12.**Graphs of the mass of ${\eta}_{h}^{n}$ with respect to time $(t={t}^{n})$ for Example 4 with the following four settings; no transmission boundary, i.e., ${\Gamma}_{\mathrm{T}}=\varnothing $ (purple), one transmission boundary, i.e., ${\Gamma}_{\mathrm{T}}={\Gamma}_{\mathrm{T}2}$ (green), two transmission boundaries, i.e., ${\Gamma}_{\mathrm{T}}={\Gamma}_{\mathrm{T}1}\cup {\Gamma}_{\mathrm{T}3}$ (blue), and three transmission boundaries, i.e., ${\Gamma}_{\mathrm{T}}={\Gamma}_{\mathrm{T}1}\cup {\Gamma}_{\mathrm{T}2}\cup {\Gamma}_{\mathrm{T}3}$ (yellow).

**Figure 13.**Contour plot of ${\eta}_{h}^{n}$ by LG2 with $\Gamma ={\overline{\Gamma}}_{\mathrm{D}}\cup {\overline{\Gamma}}_{\mathrm{T}}$ for the extended domain (

**left**) and for the original domain (

**right**) on the Bay of Bengal for $t=0,2500$ and $3000$. The green dotted lines in the left figures indicate the position of the bottom boundary of the original domain.

**Figure 14.**Contour plot of ${\eta}_{h}^{n}$ by LG2 with $\Gamma ={\overline{\Gamma}}_{\mathrm{D}}\cup {\overline{\Gamma}}_{\mathrm{T}}$ for the extended domain (

**left**) and for the original domain (

**right**) on the Bay of Bengal for $t=3500,4000$ and $5000$.

**Table 1.**Values of ${E}_{i}\left(\eta \right)$ and ${E}_{i}\left(u\right)$, $i=0,1$, by schemes LG1 and LG2 for Example 1 ($\Gamma ={\Gamma}_{\mathrm{D}}$).

LG1 | |||||
---|---|---|---|---|---|

N | $\Delta \mathit{t}$ | ${\mathit{E}}_{0}\left(\mathit{\eta}\right)$ | EOC | ${\mathit{E}}_{0}\left(\mathit{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $3.89\times {10}^{0}$ | - | $3.78\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $2.20\times {10}^{0}$ | 1.65 | $2.28\times {10}^{-2}$ | 1.45 |

32 | $4.42\times {10}^{-2}$ | $1.45\times {10}^{0}$ | 1.19 | $1.57\times {10}^{-2}$ | 1.09 |

64 | $3.13\times {10}^{-2}$ | $1.01\times {10}^{0}$ | 1.05 | $1.10\times {10}^{-2}$ | 1.03 |

128 | $2.21\times {10}^{-2}$ | $7.11\times {10}^{-1}$ | 1.01 | $7.77\times {10}^{-3}$ | 1.00 |

256 | $1.56\times {10}^{-2}$ | $5.02\times {10}^{-1}$ | 1.00 | $5.51\times {10}^{-3}$ | 0.99 |

LG1 | |||||

$\mathit{N}$ | $\Delta \mathit{t}$ | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{\eta}\right)$ | EOC | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $3.00\times {10}^{0}$ | - | $7.78\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $1.73\times {10}^{0}$ | 1.59 | $4.63\times {10}^{-2}$ | 1.49 |

32 | $4.42\times {10}^{-2}$ | $1.25\times {10}^{0}$ | 0.93 | $2.95\times {10}^{-2}$ | 1.31 |

64 | $3.13\times {10}^{-2}$ | $9.78\times {10}^{-1}$ | 0.71 | $2.04\times {10}^{-2}$ | 1.06 |

128 | $2.21\times {10}^{-2}$ | $6.42\times {10}^{-1}$ | 1.22 | $1.42\times {10}^{-2}$ | 1.04 |

256 | $1.56\times {10}^{-2}$ | $4.35\times {10}^{-1}$ | 1.12 | $1.00\times {10}^{-2}$ | 1.01 |

LG2 | |||||

$\mathit{N}$ | $\Delta \mathit{t}$ | ${\mathit{E}}_{\mathbf{0}}\left(\mathbf{\eta}\right)$ | EOC | ${\mathit{E}}_{\mathbf{0}}\left(\mathbf{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $6.81\times {10}^{-1}$ | - | $1.71\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $1.96\times {10}^{-1}$ | 3.60 | $7.03\times {10}^{-3}$ | 2.57 |

32 | $4.42\times {10}^{-2}$ | $8.53\times {10}^{-2}$ | 2.40 | $3.32\times {10}^{-3}$ | 2.16 |

64 | $3.13\times {10}^{-2}$ | $3.82\times {10}^{-2}$ | 2.32 | $1.64\times {10}^{-3}$ | 2.04 |

128 | $2.21\times {10}^{-2}$ | $1.87\times {10}^{-2}$ | 2.05 | $8.20\times {10}^{-4}$ | 1.99 |

256 | $1.56\times {10}^{-2}$ | $9.46\times {10}^{-3}$ | 1.97 | $4.17\times {10}^{-4}$ | 1.95 |

LG2 | |||||

$\mathit{N}$ | $\Delta \mathit{t}$ | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{\eta}\right)$ | EOC | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $3.97\times {10}^{0}$ | - | $5.68\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $2.24\times {10}^{0}$ | 1.65 | $2.90\times {10}^{-2}$ | 1.94 |

32 | $4.42\times {10}^{-2}$ | $2.00\times {10}^{0}$ | 0.33 | $1.20\times {10}^{-2}$ | 2.54 |

64 | $3.13\times {10}^{-2}$ | $1.64\times {10}^{0}$ | 0.57 | $6.72\times {10}^{-3}$ | 1.67 |

128 | $2.21\times {10}^{-2}$ | $1.17\times {10}^{0}$ | 0.97 | $3.23\times {10}^{-3}$ | 2.11 |

256 | $1.56\times {10}^{-2}$ | $8.64\times {10}^{-1}$ | 0.88 | $1.47\times {10}^{-3}$ | 2.28 |

**Table 2.**Values of ${E}_{i}\left(\eta \right)$ and ${E}_{i}\left(u\right)$, $i=0,1$, by schemes LG1 and LG2 for Example 2 ($\Gamma ={\overline{\Gamma}}_{\mathrm{D}}\cup {\overline{\Gamma}}_{\mathrm{T}}$).

LG1 | |||||
---|---|---|---|---|---|

N | $\Delta \mathit{t}$ | ${\mathit{E}}_{0}\left(\mathit{\eta}\right)$ | EOC | ${\mathit{E}}_{0}\left(\mathit{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $3.88\times {10}^{0}$ | - | $3.86\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $2.19\times {10}^{0}$ | 1.65 | $2.33\times {10}^{-2}$ | 1.46 |

32 | $4.42\times {10}^{-2}$ | $1.45\times {10}^{0}$ | 1.19 | $1.58\times {10}^{-2}$ | 1.11 |

64 | $3.13\times {10}^{-2}$ | $1.01\times {10}^{0}$ | 1.05 | $1.11\times {10}^{-2}$ | 1.03 |

128 | $2.21\times {10}^{-2}$ | $7.09\times {10}^{-1}$ | 1.01 | $7.82\times {10}^{-3}$ | 1.01 |

256 | $1.56\times {10}^{-2}$ | $5.01\times {10}^{-1}$ | 1.00 | $5.53\times {10}^{-3}$ | 1.00 |

LG1 | |||||

$\mathit{N}$ | $\Delta \mathit{t}$ | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{\eta}\right)$ | EOC | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $2.95\times {10}^{0}$ | - | $7.80\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $1.71\times {10}^{0}$ | 1.57 | $4.64\times {10}^{-2}$ | 1.50 |

32 | $4.42\times {10}^{-2}$ | $1.24\times {10}^{0}$ | 0.94 | $2.95\times {10}^{-2}$ | 1.31 |

64 | $3.13\times {10}^{-2}$ | $9.78\times {10}^{-1}$ | 0.67 | $2.03\times {10}^{-2}$ | 1.07 |

128 | $2.21\times {10}^{-2}$ | $6.42\times {10}^{-1}$ | 1.21 | $1.41\times {10}^{-2}$ | 1.04 |

256 | $1.56\times {10}^{-2}$ | $4.34\times {10}^{-1}$ | 1.13 | $9.96\times {10}^{-3}$ | 1.01 |

LG2 | |||||

$\mathit{N}$ | $\Delta \mathit{t}$ | ${\mathit{E}}_{\mathbf{0}}\left(\mathbf{\eta}\right)$ | EOC | ${\mathit{E}}_{\mathbf{0}}\left(\mathbf{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $6.70\times {10}^{-1}$ | - | $1.75\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $1.95\times {10}^{-1}$ | 3.56 | $7.23\times {10}^{-3}$ | 2.55 |

32 | $4.42\times {10}^{-2}$ | $8.58\times {10}^{-2}$ | 2.37 | $3.37\times {10}^{-3}$ | 2.20 |

64 | $3.13\times {10}^{-2}$ | $3.97\times {10}^{-2}$ | 2.22 | $1.67\times {10}^{-3}$ | 2.03 |

128 | $2.21\times {10}^{-2}$ | $1.87\times {10}^{-2}$ | 2.17 | $8.37\times {10}^{-4}$ | 2.00 |

256 | $1.56\times {10}^{-2}$ | $9.54\times {10}^{-3}$ | 1.94 | $4.25\times {10}^{-4}$ | 1.96 |

LG2 | |||||

$\mathit{N}$ | $\Delta \mathit{t}$ | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{\eta}\right)$ | EOC | ${\mathit{E}}_{\mathbf{1}}\left(\mathbf{u}\right)$ | EOC |

8 | $8.84\times {10}^{-2}$ | $3.89\times {10}^{0}$ | - | $5.70\times {10}^{-2}$ | - |

16 | $6.25\times {10}^{-2}$ | $2.21\times {10}^{0}$ | 1.63 | $2.93\times {10}^{-2}$ | 1.92 |

32 | $4.42\times {10}^{-2}$ | $1.98\times {10}^{0}$ | 0.32 | $1.24\times {10}^{-2}$ | 2.49 |

64 | $3.13\times {10}^{-2}$ | $1.65\times {10}^{0}$ | 0.54 | $6.90\times {10}^{-3}$ | 1.69 |

128 | $2.21\times {10}^{-2}$ | $1.17\times {10}^{0}$ | 0.97 | $3.26\times {10}^{-3}$ | 2.16 |

256 | $1.56\times {10}^{-2}$ | $8.62\times {10}^{-1}$ | 0.89 | $1.48\times {10}^{-3}$ | 2.27 |

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## Share and Cite

**MDPI and ACS Style**

Rasid, M.M.; Kimura, M.; Murshed, M.M.; Wijayanti, E.R.; Notsu, H. A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography. *Mathematics* **2023**, *11*, 1633.
https://doi.org/10.3390/math11071633

**AMA Style**

Rasid MM, Kimura M, Murshed MM, Wijayanti ER, Notsu H. A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography. *Mathematics*. 2023; 11(7):1633.
https://doi.org/10.3390/math11071633

**Chicago/Turabian Style**

Rasid, Md Mamunur, Masato Kimura, Md Masum Murshed, Erny Rahayu Wijayanti, and Hirofumi Notsu. 2023. "A Two-Step Lagrange–Galerkin Scheme for the Shallow Water Equations with a Transmission Boundary Condition and Its Application to the Bay of Bengal Region—Part I: Flat Bottom Topography" *Mathematics* 11, no. 7: 1633.
https://doi.org/10.3390/math11071633