Implementation of method of lines to predict water levels due to a storm along the coastal region of Bangladesh

Abstract

In this study, the method of lines (MOL) has been applied to solve two-dimensional vertically integrated shallow water equations in Cartesian coordinates for the prediction of water levels due to a storm surge along the coast of Bangladesh. In doing so, the partial derivatives with respect to the space variables were discretized by the finite difference (central) method to obtain a system of ordinary differential equations (ODEs) with time as independent variable. The classical fourth-order Runge–Kutta method was used to solve the obtained system of the ODEs. We used a nested finite difference scheme, where a high resolution fine grid model (FGM) capable of incorporating all major islands along the coastal region of Bangladesh was nested into a coarse grid model (CGM) covering up to 15°N latitude of the Bay of Bengal. The boundaries of the coast and islands were approximated through proper stair step. Appropriate tidal condition over the model domain was generated by forcing the sea level to be oscillatory with the constituent M ₂ along the southern open boundary of the CGM omitting wind stress. Along the northeast corner of the FGM, the Meghna River discharge was taken into account. The developed model was applied to estimate water levels along the coast of Bangladesh due to the interaction of tide and surge associated with the April 1991 storm. We also computed our results employing the standard finite difference method (FDM). Simulated results show the MOL performs well in comparison with the FDM with regard to CPU time and stability, and ensures conformity with observations.

Appendix

If any dependent variable χ(x, y, t) at a grid point (x _i , y _j ) at time t _k is represented by

$$ \chi \left( {x_{i} ,y_{j} ,t_{k} } \right) = \chi_{i,j}^{k} ,\quad \frac{1}{2}\left( {\chi_{i + 1,j}^{k} + \chi_{i - 1,j}^{k} } \right) = \overline{{\chi_{i,j}^{k} }}^{x} , $$

$$ \frac{1}{2}\left( {\chi_{i,j + 1}^{k} + \chi_{i,j - 1}^{k} } \right) = \overline{{\chi_{i,j}^{k} }}^{y} ,\,\frac{1}{4}\left( {\chi_{i + 1,j}^{k} + \chi_{i - 1,j}^{k} + \chi_{i,j + 1}^{k} + \chi_{i,j - 1}^{k} } \right) = \overline{{\chi_{i,j}^{k} }}^{xy} , $$

then Eqs. (1)–(3) after discretization can be written respectively as follows

$$ \left( {\frac{\partial \zeta }{\partial t}} \right)_{i,j} = {\text{CR}}1 + {\text{CR}}2, $$

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where i = 2, 4,…, M − 2 and j = 3, 5,…, N − 2.

$$ \left( {\frac{\partial u}{\partial t}} \right)_{i,j} = {\text{UR}}1 + {\text{UR}}2 + {\text{UR}}3 + {\text{UR}}4 + {\text{UR}}5 + {\text{UR}}6, $$

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where i = 3, 5,…, M − 1 and j = 3, 5,…, N − 2.

$$ {\text{and}}\;\left( {\frac{\partial v}{\partial t}} \right)_{i,j} = {\text{VR}}1 + {\text{VR}}2 + {\text{VR}}3 + {\text{VR}}4 + {\text{VR}}5 + {\text{VR}}6, $$

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where i = 2, 4,…, M − 2 and j = 2, 4,…, N − 1.

In Eqs. (16)–(18),

$$ {\text{CR}}1 = - \frac{{\left( {\overline{{\zeta_{i + 1,j}^{k} }}^{x} + h_{i + 1,j} } \right)u_{i + 1,j}^{k} - \left( {\overline{{\zeta_{i - 1,j}^{k} }}^{x} + h_{i - 1,j} } \right)u_{i - 1,j}^{k} }}{2\varDelta x}, $$

$$ {\text{CR}}2 = - \frac{{\left( {\overline{{\zeta_{i,j + 1}^{k} }}^{y} + h_{i,j + 1} } \right)v_{i,j + 1}^{k} - \left( {\overline{{\zeta_{i,j - 1}^{k} }}^{y} + h_{i,j - 1} } \right)v_{i,j - 1}^{k} }}{2\varDelta y}, $$

$$ {\text{UR}}1 = - \left\{ \begin{gathered} u_{i,j}^{k} \frac{{u_{i + 2,j}^{k} - u_{i - 2,j}^{k} }}{4\varDelta x},\quad \quad {\text{for}}\;i \ne M - 1 \hfill \\ u_{i,j}^{k} \frac{{0.5\left( {3u_{i,j}^{k} - u_{i - 2,j}^{k} } \right) - u_{i - 2,j}^{k} }}{4\varDelta x},\quad {\text{for}}\;i = M - 1 \hfill \\ \end{gathered} \right. $$

$$ {\text{UR}}2 = - \overline{{v_{i,j}^{k} }}^{xy} \frac{{\overline{{u_{i,j + 1}^{k} }}^{y} - \overline{{u_{i,j - 1}^{k} }}^{y} }}{2\varDelta y},\quad {\text{UR}}3 = f_{i,j} \overline{{v_{i,j}^{k} }}^{xy} ,\quad {\text{UR}}4 = - g\frac{{\zeta_{i + 1,j}^{k + 1} - \zeta_{i - 1,j}^{k + 1} }}{2\varDelta x}, $$

$$ {\text{UR}}5 = \frac{{T_{x} }}{{\rho \left( {\overline{{\zeta_{i,j}^{k + 1} }}^{x} + h_{i,j} } \right)}},\quad {\text{UR}}6 = - \frac{{C_{f} u_{i,j}^{k} }}{{\overline{{\zeta_{i,j}^{k + 1} }}^{x} + h_{i,j} }}\left[ {\left( {u_{i,j}^{k} } \right)^{2} + \left( {\overline{{v_{i,j}^{k} }}^{xy} } \right)^{2} } \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} , $$

$$ {\text{VR}}1 = - \left\{ \begin{gathered} \overline{{u_{i,j}^{k} }}^{xy} \frac{{\overline{{v_{i + 1,j}^{k} }}^{x} - \overline{{v_{i - 1,j}^{k} }}^{x} }}{2\varDelta x},\quad \quad {\text{for}}\;i \ne 2 \hfill \\ \overline{{u_{i,j}^{k} }}^{xy} \frac{{\overline{{v_{i + 1,j}^{k} }}^{x} - 0.5\left( {3v_{i,j}^{k} - v_{i + 2,j}^{k} } \right)}}{2\varDelta x},\quad {\text{for}}\;i = 2 \hfill \\ \end{gathered} \right. $$

$$ {\text{VR}}2 = - \left\{ \begin{gathered} v_{i,j}^{k} \frac{{v_{i,j + 2}^{k} - v_{i,j - 2}^{k} }}{4\varDelta y},\quad \quad {\text{for}}\;j \ne 2,\;j \ne N - 1 \hfill \\ v_{i,j}^{k} \frac{{v_{i,j + 2}^{k} - 0.5\left( {3v_{i,j}^{k} - v_{i,j + 2}^{k} } \right)}}{4\varDelta y},\quad {\text{for}}\;j = 2 \hfill \\ v_{i,j}^{k} \frac{{0.5\left( {3v_{i,j}^{k} - v_{i,j - 2}^{k} } \right) - v_{i,j - 2}^{k} }}{4\varDelta y},\quad {\text{for}}\;j = N - 1 \hfill \\ \end{gathered} \right. $$

$$ {\text{VR}}3 = - f_{i,j} \overline{{u_{i,j}^{k} }}^{xy} ,\quad {\text{VR}}4 = - g\frac{{\zeta_{i,j + 1}^{k + 1} - \zeta_{i,j - 1}^{k + 1} }}{2\varDelta y},\quad {\text{VR}}5 = \frac{{T_{y} }}{{\rho \left( {\overline{{\zeta_{i,j}^{k + 1} }}^{y} + h_{i,j} } \right)}}\;{\text{and}} $$

$$ {\text{VR}}6 = - \frac{{C_{f} v_{i,j}^{k} }}{{\overline{{\zeta_{i,j}^{k + 1} }}^{y} + h_{i,j} }}\left[ {\left( {\overline{{u_{i,j}^{k} }}^{xy} } \right)^{2} + \left( {v_{i,j}^{k} } \right)^{2} } \right]^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} . $$

From the boundary conditions, given by Eqs. (6)–(8), the elevations at j = 1, j = N and i = M are computed respectively in the following manner.

$$ \zeta_{i,1}^{k + 1} = - \zeta_{i,3}^{k + 1} - 2\sqrt {\left( {h_{i,2} /g} \right)} \;v_{i,2}^{k} , $$

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$$ \zeta_{i,N}^{k + 1} = - \zeta_{i,N - 2}^{k + 1} + 2\sqrt {\left( {h_{i,N - 1} /g} \right)} \;v_{i,N - 1}^{k} , $$

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$$ \zeta_{M,j}^{k + 1} = - \zeta_{M - 2,j}^{k + 1} + 2\sqrt {\left( {{{h_{M - 1,j} } \mathord{\left/ {\vphantom {{h_{M - 1,j} } g}} \right. \kern-0pt} g}} \right)} u_{M - 1,j}^{k} + 4a\sin \left( {{{2\pi t} \mathord{\left/ {\vphantom {{2\pi t} {T + \varphi }}} \right. \kern-0pt} {T + \varphi }}} \right), $$

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where i = 2, 4, 6,…, M − 2 and j = 1, 3, 5, 7,…, N.