Parallel Implementation of Dispersive Tsunami Wave Modeling with a Nesting Algorithm for the 2011 Tohoku Tsunami

Abstract

Because of improvements in offshore tsunami observation technology, dispersion phenomena during tsunami propagation have often been observed in recent tsunamis, for example the 2004 Indian Ocean and 2011 Tohoku tsunamis. The dispersive propagation of tsunamis can be simulated by use of the Boussinesq model, but the model demands many computational resources. However, rapid progress has been made in parallel computing technology. In this study, we investigated a parallelized approach for dispersive tsunami wave modeling. Our new parallel software solves the nonlinear Boussinesq dispersive equations in spherical coordinates. A variable nested algorithm was used to increase spatial resolution in the target region. The software can also be used to predict tsunami inundation on land. We used the dispersive tsunami model to simulate the 2011 Tohoku earthquake on the Supercomputer K. Good agreement was apparent between the dispersive wave model results and the tsunami waveforms observed offshore. The finest bathymetric grid interval was 2/9 arcsec (approx. 5 m) along longitude and latitude lines. Use of this grid simulated tsunami soliton fission near the Sendai coast. Incorporating the three-dimensional shape of buildings and structures led to improved modeling of tsunami inundation.

Appendix: Finite-difference Scheme for Nonlinear Dispersive Equations

The finite difference calculation was performed in the staggered-grid system shown in Fig. 15. Because the integration over time was solved with a leapfrog method, the water height (h) was defined at time $t = n\Delta t$, and the depth-integrated quantities (M, N) were defined at $t = (n - 1/2)\Delta t$, where $\Delta t$ is the time step and $n = 1,2,3 \ldots$. In the Appendix, φ and θ indicate the longitude and co-latitude, respectively, R is the earth’s radius, d is the water depth, and D is total depth, that is d + h. g is the gravitational constant, f is the Coriolis parameter, and n is the Manning’s roughness coefficient. We considered the finite-difference form of the dispersion term, the final term on the right-hand side of Eq. (1).

Fig. 15

Staggered-grid system for the finite-difference simulation of nonlinear dispersive equations

$$ \frac{{d^{2} }}{3R\;\sin \theta }\frac{\partial }{\partial \varphi }\left[ {\frac{1}{R\;\sin \theta }\left( {\frac{{\partial^{2} M}}{\partial \varphi \partial t} + \frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \theta \partial t}} \right)} \right] = \frac{{d^{2} }}{{3R^{2} \sin^{2} \theta }}\frac{\partial }{\partial t}\left( {\frac{{\partial^{2} M}}{{\partial \varphi^{2} }} + \frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \varphi \partial \theta }} \right), $$

(4)

where

$$ \frac{{\partial^{2} M}}{{\partial \varphi^{2} }} = \frac{{M_{i + 1,j} - 2M_{i,j} + M_{i - 1,j} }}{{\Delta \varphi^{2} }}, $$

(5)

$\Delta \varphi$ is the grid size along a longitude line. We introduced

$$ U = M_{i + 1,j} - 2M_{i,j} + M_{i - 1,j} , $$

(6)

to write

$$ \frac{\partial }{\partial t}\left( {\frac{{\partial^{2} M}}{{\partial \varphi^{2} }}} \right) = \frac{\partial }{\partial t}\left( {\frac{U}{{\Delta \varphi^{2} }}} \right) = \frac{{U^{{n + \frac{1}{2}}} - U^{{n - \frac{1}{2}}} }}{{\Delta \varphi^{2} \Delta t}}. $$

(7)

The other term of Eq. (4) can be expressed as:

$$ \begin{gathered} \frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \varphi \partial \theta } \; = \; \frac{\partial }{\partial \varphi }\left( {\frac{{\bar{N}_{i,j + 1} \sin \left( {\theta + \Delta \theta } \right) - \bar{N}_{i,j} \sin \left( \theta \right)}}{\Delta \theta }} \right) \\ \, = \, \frac{{\bar{N}_{i,j + 1} \sin \left( {\theta + \Delta \theta } \right) - \bar{N}_{i - 1,j + 1} \sin \left( {\theta + \Delta \theta } \right) - \bar{N}_{i,j} \sin \left( \theta \right) + \bar{N}_{i - 1,j} \sin \left( \theta \right)}}{\Delta \theta \Delta \varphi }, \\ \end{gathered} $$

(8)

where $\Delta \theta$ is the grid size along a latitude line. Because of the staggered grid system:

$$ \bar{N}_{i,j} = \frac{{N_{i,j} + N_{i + 1,j} + N_{i,j - 1} + N_{i + 1,j - 1} }}{4}. $$

(9)

By defining:

$$ \bar{V} = \bar{N}_{i,j + 1} \sin \left( {\theta + \Delta \theta } \right) - \bar{N}_{i - 1,j + 1} \sin \left( {\theta + \Delta \theta } \right) - \bar{N}_{i,j} \sin \left( \theta \right) + - \bar{N}_{i - 1,j} \sin \left( \theta \right), $$

(10)

we can write:

$$ \frac{\partial }{\partial t}\left( {\frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \varphi \partial \theta }} \right) = \frac{\partial }{\partial t}\left( {\frac{{\bar{V}}}{\Delta \theta \Delta \varphi }} \right) = \frac{{\bar{V}^{{n + \frac{1}{2}}} - \bar{V}^{{n - \frac{1}{2}}} }}{\Delta \theta \Delta \varphi \Delta t}. $$

(11)

Accordingly, the dispersion term of Eq. (1) can be expressed by using Eqs. (7) and (11) in finite-difference form as:

$$ \frac{{d^{2} }}{3R\;\sin \theta }\frac{\partial }{\partial \varphi }\left[ {\frac{1}{R\;\sin \theta }\left( {\frac{{\partial^{2} M}}{\partial \varphi \partial t} + \frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \theta \partial t}} \right)} \right] = \frac{{\bar{d}_{i,j}^{2} }}{{3R^{2} \sin^{2} \theta \Delta t}}\left( {\frac{{U^{{n + \frac{1}{2}}} - U^{{n - \frac{1}{2}}} }}{{\Delta \varphi^{2} }} + \frac{{\bar{V}^{{n + \frac{1}{2}}} - \bar{V}^{{n - \frac{1}{2}}} }}{\Delta \theta \Delta \varphi }} \right), $$

(12)

where also, because of the staggered grid system:

$$ \bar{d}_{i,j} = \frac{{d_{i,j} + d_{i + 1,j} }}{2} $$

(13)

Finally, Eq. (1) can be written in finite-difference form as:

$$ M^{{n + \frac{1}{2}}} = M^{{n - \frac{1}{2}}} - \frac{1}{R\;\sin \theta }\frac{\Delta t}{\Delta \varphi }\left( {\frac{{M_{i + 1,j}^{{n - \frac{1}{2}2}} }}{{{\bar{\text{D}}}_{i + 1,j} }} - \frac{{M_{i,j}^{{n - \frac{1}{2}2}} }}{{{\bar{\text{D}}}_{i,j} }}} \right)\;\; - \frac{1}{R}\frac{\Delta t}{\Delta \theta }\left( {\frac{{M_{i,j + 1}^{{n - \frac{1}{2}}} \bar{N}_{i,j + 1}^{{n - \frac{1}{2}}} }}{{{\bar{\text{D}}}_{i,j + 1} }} - \frac{{M_{i,j}^{{n - \frac{1}{2}}} \bar{N}_{i,j}^{{n - \frac{1}{2}}} }}{{{\bar{\text{D}}}_{i,j} }}} \right) + \frac{{g{\bar{\text{D}}}_{i,j} }}{R\;\sin \theta }\frac{\Delta t}{\Delta \varphi }\left( {h_{i + 1,j}^{n} - h_{i,j}^{n} } \right)\;\; - f\bar{N}_{i,j}^{{n - \frac{1}{2}}} \Delta t - \frac{{gn^{2} }}{{\bar{D}_{i,j}^{{{\raise0.7ex\hbox{$7$} \!\mathord{\left/ {\vphantom {7 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}M_{i,j}^{{n - \frac{1}{2}}} \sqrt {M_{i,j}^{{n - \frac{1}{2}2}} + \bar{N}_{i,j}^{{n - \frac{1}{2}2}} } \Delta t\;\; + \frac{{\bar{d}_{i,j}^{2} }}{{3R^{2} \sin^{2} \theta }}\left( {\frac{{U^{{n + \frac{1}{2}}} - U^{{n - \frac{1}{2}}} }}{{\Delta \varphi^{2} }} + \frac{{\bar{V}^{{n + \frac{1}{2}}} - \bar{V}^{{n - \frac{1}{2}}} }}{\Delta \theta \Delta \varphi }} \right), $$

(14)

where, again, because of the staggered grid system:

$$ {\bar{\text{D}}}_{i,j} = \frac{{d_{i,j} + d_{i + 1,j} + h_{i,j} + h_{i + 1,j} }}{2} $$

(15)

It should be noted that the 2nd and 3rd terms on the right-hand side of Eq. (14) were approximated with upwind finite differences.

Similarly, we considered the finite difference form for the dispersion term (the final term on the left-hand side) of Eq. (2):

$$ \frac{{d^{2} }}{3R}\frac{\partial }{\partial \theta }\left[ {\frac{1}{R\;\sin \theta }\left( {\frac{{\partial^{2} M}}{\partial \varphi \partial t} + \frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \theta \partial t}} \right)} \right] = \frac{{d^{2} }}{{3R^{2} }}\frac{\partial }{\partial t}\left[ {\frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial M}{\partial \varphi }} \right) + \frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial N\;\sin \theta }{\partial \theta }} \right)} \right] $$

(16)

where:

$$ \frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial M}{\partial \varphi }} \right) = \frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{{\bar{M}_{i + 1,j} - \bar{M}_{i,j} }}{\Delta \varphi }} \right) = \frac{1}{\Delta \theta \Delta \varphi }\left( {\frac{{\bar{M}_{i + 1,j} - \bar{M}_{i,j} }}{\sin \theta } - \frac{{\bar{M}_{i + 1,j - 1} - \bar{M}_{i,j - 1} }}{\sin (\theta - \Delta \theta )}} \right), $$

(17)

and because of the staggered grid system:

$$ \bar{M}_{i,j} = \frac{{M_{i,j} + M_{i,j + 1} + M_{i - 1,j} + M_{i - 1,j + 1} }}{4}. $$

(18)

By further defining:

$$ \bar{U} = \frac{{\bar{M}_{i + 1,j} - \bar{M}_{i,j} }}{\sin \theta } - \frac{{\bar{M}_{i + 1,j - 1} - \bar{M}_{i,j - 1} }}{\sin (\theta - \Delta \theta )}, $$

(19)

we can write:

$$ \frac{\partial }{\partial t}\left[ {\frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial M}{\partial \varphi }} \right)} \right] = \frac{\partial }{\partial t}\left( {\frac{{\bar{U}}}{\Delta \theta \Delta \varphi }} \right) = \frac{{\bar{U}^{{n + \frac{1}{2}}} - \bar{U}^{{n - \frac{1}{2}}} }}{\Delta \theta \Delta \varphi \Delta t}. $$

(20)

The other term of Eq. (16) can be expressed as:

$$ \begin{gathered} \frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial N\;\sin \theta }{\partial \theta }} \right) = \frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{{N_{{i,j + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \sin \left( {\theta + {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) - N_{{i,j - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \sin \left( {\theta - {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}{\Delta \theta }} \right) \\ = \frac{1}{\Delta \theta }\frac{\partial }{\partial \theta }\left( {\frac{{N_{{i,j + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \sin \left( {\theta + {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) - N_{{i,j - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} \sin \left( {\theta - {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}{\sin \theta }} \right) \\ = \frac{1}{{\Delta \theta^{2} }}\left( {\frac{{N_{i,j + 1} \sin \left( {\theta + \Delta \theta } \right) - N_{i,j} \sin \left( \theta \right)}}{{\sin \left( {\theta + {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}} - \frac{{N_{i,j} \sin \left( \theta \right) - N_{i,j - 1} \sin \left( {\theta - \Delta \theta } \right)}}{{\sin \left( {\theta - {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}} \right). \\ \end{gathered} $$

(21)

By defining:

$$ V = \frac{{N_{i,j + 1} \sin \left( {\theta + \Delta \theta } \right) - N_{i,j} \sin \left( \theta \right)}}{{\sin \left( {\theta + {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}} - \frac{{N_{i,j} \sin \left( \theta \right) - N_{i,j - 1} \sin \left( {\theta - \Delta \theta } \right)}}{{\sin \left( {\theta - {\raise0.7ex\hbox{${\Delta \theta }$} \!\mathord{\left/ {\vphantom {{\Delta \theta } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right)}}, $$

(22)

and we can write:

$$ \frac{\partial }{\partial t}\left( {\frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial N\;\sin \theta }{\partial \theta }} \right)} \right) = \frac{1}{{\Delta \theta^{2} }}\frac{\partial V}{\partial t} = \frac{{V^{{n + \frac{1}{2}}} - V^{{n - \frac{1}{2}}} }}{{\Delta t\Delta \theta^{2} }}. $$

(23)

By using Eqs. (20) and (23), the dispersion term of Eq. (2) can be expressed in finite-difference form as:

$$ \frac{{d^{2} }}{3R}\frac{\partial }{\partial \theta }\left[ {\frac{1}{R\;\sin \theta }\left( {\frac{{\partial^{2} M}}{\partial \varphi \partial t} + \frac{{\partial^{2} \left( {N\;\sin \theta } \right)}}{\partial \theta \partial t}} \right)} \right] = \frac{{\bar{d}_{i,j}^{2} }}{3R}\left( {\frac{{\bar{U}^{{n + \frac{1}{2}}} - \bar{U}^{{n - \frac{1}{2}}} }}{\Delta \theta \Delta \varphi \Delta t} + \frac{{V^{{n + \frac{1}{2}}} - V^{{n - \frac{1}{2}}} }}{{\Delta t\Delta \theta^{2} }}} \right), $$

(24)

where because of the staggered grid system:

$$ \bar{d}_{i,j} = \frac{{d_{i,j} + d_{i,j + 1} }}{2}. $$

(25)

Finally, Eq. (2) can be written in finite-difference form as:

$$ \begin{gathered} N^{{n + \frac{1}{2}}} = N^{{n - \frac{1}{2}}} - \frac{1}{R\;\sin \theta }\frac{\Delta t}{\Delta \varphi }\left( {\frac{{\bar{M}_{i,j + 1}^{{n - \frac{1}{2}}} N_{i,j + 1}^{{n - \frac{1}{2}}} }}{{{\bar{\text{D}}}_{i,j + 1} }} - \frac{{\bar{M}_{i,j}^{{n - \frac{1}{2}}} N_{i,j}^{{n - \frac{1}{2}}} }}{{{\bar{\text{D}}}_{i,j} }}} \right) - \frac{1}{R}\frac{\Delta t}{\Delta \theta }\left( {\frac{{N_{i + 1,j}^{{n - \frac{1}{2}2}} }}{{{\bar{\text{D}}}_{i + 1,j} }} - \frac{{N_{i,j}^{{n - \frac{1}{2}2}} }}{{{\bar{\text{D}}}_{i,j} }}} \right) \hfill \\ + \frac{{g{\bar{\text{D}}}_{i,j} }}{R}\frac{\Delta t}{\Delta \theta }\left( {h_{i,j + 1}^{n} - h_{i,j}^{n} } \right) + f\bar{M}_{i,j}^{{n - \frac{1}{2}}} \Delta t \hfill \\ - \frac{{gn^{2} }}{{\bar{D}_{i,j}^{{{\raise0.7ex\hbox{$7$} \!\mathord{\left/ {\vphantom {7 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}N_{i,j}^{{n - \frac{1}{2}}} \sqrt {\bar{M}_{i,j}^{{n - \frac{1}{2}2}} + N_{i,j}^{{n - \frac{1}{2}2}} } \Delta t \hfill \\ + \frac{{\bar{d}_{i,j}^{2} }}{{3R^{2} }}\left( {\frac{{\bar{U}^{{n + \frac{1}{2}}} - \bar{U}^{{n - \frac{1}{2}}} }}{\Delta \theta \Delta \varphi } + \frac{{V^{{n + \frac{1}{2}}} - V^{{n - \frac{1}{2}}} }}{{\Delta \theta^{2} }}} \right), \hfill \\ \end{gathered} $$

(26)

where, because of the staggered grid system:

$$ {\bar{\text{D}}}_{i,j} = \frac{{d_{i,j} + d_{i,j + 1} + h_{i,j} + h_{i,j + 1} }}{2}. $$

(27)

It should also be noted that the 2nd and 3rd terms on the right-hand side of Eq. (26) are approximated with upwind finite differences.

Equation (3) can be written in the finite-difference form as:

$$ h^{n + 1} = h^{n} - \frac{\Delta t}{R\;\sin \theta }\left[ {\left( {\frac{{\left( {M_{i,j}^{{n + \frac{1}{2}}} - M_{i - 1,j}^{{n + \frac{1}{2}}} } \right)}}{\Delta \varphi } + \frac{{\left( {N_{i,j}^{{n + \frac{1}{2}}} \sin \left( \theta \right) - N_{i,j - 1}^{{n + \frac{1}{2}}} \sin \left( {\theta - \Delta \theta } \right)} \right)}}{\Delta \theta }} \right)} \right]. $$

(28)

At time $t = n\Delta t$, we calculate $M^{{n + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ and $N^{{n + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ by substituting $h^{n}$, $M^{{n - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$, and $N^{{n - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ into Eqs. (14) and (26). These equations are solved by use of an iterative method (Gauss–Seidel method) (Press et al. 1986). The calculated $M^{{n + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ and $N^{{n + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ are substituted into Eq. (28) to obtain $h^{n + 1}$. Then $h^{n + 1}$, $M^{{n + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$, and$N^{{n + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}$ are used to solve Eqs. (14), (26), and (28) at the next time step.