Performance Benchmarking of TsunamiHySEA Model for NTHMP’s Inundation Mapping Activities
Abstract
The TsunamiHySEA model is used to perform some of the numerical benchmark problems proposed and documented in the “Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop”. The final aim is to obtain the approval for TsunamiHySEA to be used in projects funded by the National Tsunami Hazard Mitigation Program (NTHMP). Therefore, this work contains the numerical results and comparisons for the five benchmark problems (1, 4, 6, 7, and 9) required for such aim. This set of benchmarks considers analytical, laboratory, and field data test cases. In particular, the analytical solution of a solitary wave runup on a simple beach, and its laboratory counterpart, two more laboratory tests: the runup of a solitary wave on a conically shaped island and the runup onto a complex 3D beach (Monai Valley) and, finally, a field data benchmark based on data from the 1993 Hokkaido NanseiOki tsunami.
Keywords
Numerical modeling model benchmarking tsunami HySEA model inundation1 Introduction
According to the 2006 Tsunami Warning and Education Act, all inundation models used in National Tsunami Hazard Mitigation Program (NTHMP) projects must meet benchmarking standards and be approved by the NTHMP Mapping and Modeling Subcommittee (MMS). To this end, a workshop was held in 2011 by the MMS, and participating models whose results were approved for tsunami inundation modeling were documented in the “Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop” (NTHMP 2012). Since then, other models have been subjected to the benchmark problems used in the workshop, and their approval and use subsequently requested for NTHMP projects. For those currently wishing to benchmark their tsunami inundation models, a first step consists of completing benchmark problems 1, 4, 6, 7, and 9 in NTHMP (2012). This is the aim of the present benchmarking study for the case of the TsunamiHySEA model. Another preliminary requirement for achieving MMS approval for tsunami inundation models is that all models being used by US federal, state, territory, and commonwealth governments should be provided to the public as open source. A freely accessible open source version of TsunamiHySEA can be downloaded from the website https://edanya.uma.es/hysea.
Besides NTHMP (2012) and references therein, for NTHMPbenchmarked tsunami models, other authors have performed similar benchmarking efforts as the one presented here with their particular models, as is the case of Nicolsky et al. (2011), Apotsos et al. (2011) or Tolkova (2014). In addition, a model intercomparison of eight NTHMP models for benchmarks 4 (laboratory simple beach) and 6 (conical island) can be found in the study by Horrillo et al. (2015).
2 The TsunamiHySEA Model
HySEA (Hyperbolic Systems and Efficient Algorithms) software consists of a family of geophysical codes based on either singlelayer, twolayer stratified systems or multilayer shallowwater models. HySEA codes have been developed by EDANYA Group (https://edanya.uma.es) from the Universidad de Málaga (UMA) for more than a decade and they are in continuous evolution and upgrading. TsunamiHySEA is the numerical model specifically designed for tsunami simulations. It combines robustness, reliability, and good accuracy in a model based on a GPU faster than realtime (FTRT) implementation. It has been thoroughly tested, and in particular has passed not only all tests by Synolakis et al. (2008), but also other laboratory tests and proposed benchmark problems. Some of them can be found in the studies by Castro et al. (2005, 2006, 2012), Gallardo et al. (2007), de la Asunción et al. (2013), and NTHMP (2016).
2.1 Model Equations
In the previous set of equations, \(h\left( {\varvec{x},t} \right)\) denotes the thickness of the water layer at point \(\varvec{x} \in D \subset {\mathbb{R}}^{2}\) at time \(t\), with \(D\) being the horizontal projection of the 3D domain where tsunami takes place. \(H\left( \varvec{x} \right)\) is the depth of the bottom at point \(\varvec{x}\) measured from a fixed level of reference. \(u\left( {\varvec{x},t} \right)\) and \(v\left( {\varvec{x},t} \right)\) are the heightaveraged velocity in the x and ydirections, respectively, and g denotes gravity. Let us also define the function \(\eta \left( {\varvec{x},\varvec{t}} \right) = h\left( {\varvec{x},t} \right)  H(\varvec{x})\) that corresponds to the free surface of the fluid.
 1.The Manning law:$$S_{x} =  \frac{{ghM_{n}^{2} u\(u,v)\}}{{h^{4/3} }},$$$$S_{y} =  \frac{{ghM_{n}^{2} v\(u,v)\}}{{h^{4/3} }},$$
 2.A quadratic law:$$S_{x} =  c_{f} u \(u,v)\, S_{y} =  c_{f} v \(u,v)\,$$
where \(c_{f} > 0\) is the friction coefficient. In all the numerical tests presented in this study the Manning law is used.
The dispersive system implemented can be interpreted as a generalized Yamazaki model (Yamazaki et al. 2009) where the term \(\frac{\partial h}{\partial t}w\) is not neglected in the equation for the vertical velocity. The free divergence equation has been multiplied by \(h^{2}\) to write it with the conserved variables \(hu\) and \(hv\). In addition, due to the rewriting of the last equation, no special treatment is required in the presence of wet–dry fronts. The breaking criteria employed is similar to the criteria presented by Roeber et al. (2010), based on an “eddy viscosity” approach.
2.2 Numerical Solution Method
TsunamiHySEA solves the twodimensional shallowwater system using a highorder (second and third order) pathconservative finitevolume method. Values of \(h, hu\) and \(hv\) at each grid cell represent cell averages of the water depth and momentum components. The numerical scheme is conservative for both mass and momentum in flat bathymetries and, in general, is mass preserving for arbitrary bathymetries. High order is achieved by a nonlinear total variation diminishing (TVD) reconstruction operator of the unknowns \(h, hu, hv\) and \(\eta = h  H\). Then, the reconstruction of \(H\) is recovered using the reconstruction of \(h\) and \(\eta\). Moreover, in the reconstruction procedure, the positivity of the water depth is ensured. TsunamiHySEA implements several reconstruction operators: MUSCL (Monotonic UpstreamCentered Scheme for Conservation Laws, see van Leer 1979) that achieves second order, the hyperbolic Marquina’s reconstruction (see Marquina 1994) that achieves third order, and a TVD combination of piecewise parabolic and linear 2D reconstructions that also achieves third order [see Gallardo et al. (2011)]. The highorder time discretization is performed using the second or thirdorder TVD Runge–Kutta method described in Gottlieb and Shu (1998). At each cell interface, TsunamiHySEA uses Godunov’s method based on the approximation of 1D projected Riemann problems along the normal direction to each edge. In particular TsunamiHySEA implements a PVMtype (polynomial viscosity matrix) method that uses the fastest and the slowest wave speeds, similar to HLL (Harten–Lax–van Leer) method (see Castro and FernándezNieto 2012). A general overview of the derivation of the highorder methods is shown by Castro et al. 2009. For large computational domains and in the framework of Tsunami Early Warning Systems, TsunamiHySEA also implements a twostep scheme similar to leapfrog for the deepwater propagation step and a secondorder TVDweighted averaged flux (WAF) fluxlimiter scheme, described by de la Asunción et al. 2013, for close to coast propagation/inundation step. The combination of both schemes guaranties the mass conservation in the complete domain and prevents the generation of spurious highfrequency oscillations near discontinuities generated by leapfrog type schemes. At the same time, this numerical scheme reduces computational times compared with other numerical schemes, while the amplitude of the first tsunami wave is preserved.
Concerning the wet–dry fronts discretization, TsunamiHySEA implements the numerical treatment described by Castro et al. (2005) and Gallardo et al. (2007) that consists of locally replacing the 1D Riemann solver used during the propagation step, by another 1D Riemann solver that takes into account the presence of a dry cell. Moreover, the reconstruction step is also modified to preserve the positivity of the water depth. The resulting schemes are well balanced for the water at rest, that is, they exactly preserve the water at rest solutions, and are second or thirdorder accurate, depending on the reconstruction operator and the time stepping method. Finally, the numerical implementation of TsunamiHySEA has been performed on GPU clusters (de la Asunción et al. 2011, 2013, Castro et al. 2011) and nestedgrids configurations are available (Macías et al. 2013, 2014, 2015, 2016). These facts allow to speed up the computations, being able to perform complex simulations, in very large domains, much faster than real time (Macías et al. 2013, 2014, 2016).
The dispersive model implements a formal secondorder wellbalanced hybrid finitevolume/difference (FV/FD) numerical scheme. The nonhydrostatic system can be split into two parts: one corresponding to the nonlinear shallowwater component in conservative form and the other corresponding to the nonhydrostatic terms. The hyperbolic part of the system is discretized using a PVM pathconservative finitevolume method (Castro and FernándezNieto 2012 and Parés 2006), and the dispersive terms are discretized with compact finite differences. The resulting ODE system in time is discretized using a TVD Runge–Kutta method (Gottlieb and Shu 1998).
3 Benchmark Problem Comparisons
This section contains the TsunamiHySEA results for each of the five benchmark problems that are required by the NTHMP Tsunami Inundation Model Approval Process (July 2015). The specific version of TsunamiHySEA code benchmarked in the present study is the second order with MUSCL reconstruction and its secondorder dispersive counterpart when dispersion is required. Detailed descriptions of all benchmarks, as well as topography data when required and laboratory or field data for comparison when applicable, can be found in the repository of benchmark problems https://gitub.com/rjleveque/nthmpbenchmarkproblems for NTHMP, or in the NCTR repository http://nctr.pmel.noaa.gov/benchmark/. Results from model participating in original 2011 workshop can be found at NTHMP (2012). For the sake of completeness, a brief description of each benchmark problem is provided. For BP#1 and BP#4, dealing with analytical solutions or very simple laboratory 1D configurations, nondimensional variables are used everywhere. For problems dealing with 2D complex laboratory experiments (BP#6 and BP#7) scaled dimensional problems are solved. Finally, BP#9 dealing with field data is solved in realworld notscaled dimensional variables.
3.1 Benchmark Problem #1: Simple Wave on a Simple Beach—analytical—CASE H/d = 0.019
3.1.1 Problem Setup

Friction: no friction (as required).

Parameters: d = 1, g = 1, and H = 0.019 (see Fig. 1 for d and H).

Computational domain: the computational domain in x spanned from x = −10 to x = 70.

Boundary conditions: a nonreflective boundary condition at the right side of the computational domain is imposed (beach slope is located to the left).
 Initial condition: the prescribed soliton at time t = 0 with the proposed correction for the initial velocity. These initial data were given by:$$\eta \left( {x,0} \right) = H { \sec }h^{2} (\gamma (x  {\text{X}}_{ 1} )/d),$$where \(X_{1} = X_{0} + L\), with \(L = {\text{arccosh}}(\sqrt {20} )/\gamma\) the halflength of the solitary wave, and \(\gamma = \sqrt {3H/4d}\) the water wave elevation andfor the initial velocity (the minus sign meaning approaching the coast, that in the numerical test is on the lefthand side).$$u\left( {x,0} \right) =  \sqrt {\frac{g}{d}} \eta (x,0)$$

Grid resolution: the numerical results presented are for a computational mesh composed of 800 cells, i.e., Δx = 0.1 = d/10. For the convergence analysis of the maximum runup, two other increased resolutions have been used, Δx = 0.05 = d/20 and Δx = 0.025 = d/40 with 1600 and 3200 cells, respectively.

Time stepping: variable time stepping based on a CFL condition is used.

CFL: CFL number is set to 0.9.

Versions of the code: TsunamiHySEA thirdorder (with Marquina’s reconstruction) and secondorder (with MUSCL reconstruction) models have been benchmarked using this particular problem. Both models give nearly identical results.
3.1.2 Tasks to be Performed
 1.
Numerically compute the maximum runup of the solitary wave.
 2.
Compare the numerically and analytically computed water level profiles at t = 25 (d/g)^{1/2}, t = 35 (d/g)^{1/2}, t = 45 (d/g)^{1/2}, t = 55 (d/g)^{1/2}, and t = 65 (d/g)^{1/2}. Note that as we used the MATLAB scripts and data provided by Juan Horrillo on behalf of the NTHMP, the numerical vs analytical comparison is performed at the times given in the provided data and depicted by the corresponding MATLAB script that does not correspond exactly with all the time instants given in BP1 description. More precisely, they do correspond to t = 35:5:65 (d/g)^{1/2}. Therefore, t = 25 (d/g)^{1/2} is missing and t = 40, 50, and 60 (d/g)^{1/2} are shown.
 3.
Compare the numerically and analytically computed water level dynamics at locations x/d = 0.25 and x/d = 9.95 during propagation and reflection of the wave.
 4.
Demonstrate scalability of the code.
3.1.3 Numerical Results
In this section, we present the numerical results obtained using TsunamiHySEA for BP1 according to the tasks to be performed as given in the benchmark description.
3.1.3.1 Maximum Runup
The maximum runup is reached at t = 55 (d/g)^{1/2}. In the case of the reference numerical experiment with Δx = 0.1 and 800 cells, the value for the maximum runup is 0.08724. For the refined mesh experiments with Δx = 0.05 and Δx = 0.025, the computed runups are 0.09102 and 0.9165, respectively. Comparison of the numerical solutions with the analytical reference is depicted in Fig. 2 showing the convergence of the maximum runup to the analytical value as mesh size is reduced. It must be noted that for the analytical solution at time t = 55 (d/g)^{1/2} and location x = −1.8 water surface is located at 0.0909, but this is not the value of the analytical runup (that must be a value slightly above 0.92), as can be seen in Fig. 2.
Figure 3 depicts the time evolution for the maximum runup simulated for the three spatial resolutions considered. The black dot marks the approximate location of the analytical maximum runup.
3.1.3.2 Water Level at t = 35:5:65 (d/g)^{1/2}. (MATLAB Script and Data from J. Horrillo)
The next two figures show the water level profiles during the runup of the nonbreaking wave in the case H/d = 0.019 on the 1:19.85 beach at times t = 35:5:50 (d/g)^{1/2} in Fig. 4 and times t = 55:5:65 (d/g)^{1/2} in Fig. 5. For a quantitative comparison with the analytical solution, normalized root mean square deviation (NRMSD) and maximum wave amplitude error (ERR) are computed and shown for each time.
TsunamiHySEA model surface profile errors with respect to the analytical solution for H = 0.019 at times t = 35:5:65 (d/g)^{1/2}. Comparison with the mean value for NTHMP models in NTHMP (2012)
Model error for case H = 0.019  

t = 35  t = 40  t = 45  t = 50  t = 55  t = 60  t = 65  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%) 
TsunamiHySEA model error  
1  1  1  0  1  0  0  3  0  1  0  0  2  1  0.85  0.84 
Mean error for NTHMP models  
2  2  2  2  2  2  1  2  0  0  0  1  5  3  2  2 
3.1.3.3 Water Level at Locations x/d = 0.25 and x/d = 9.95
TsunamiHySEA model sea level time series errors with respect to the analytical solution for H = 0.019 at x = 9.95 and x = 0.25. Comparison with the mean value for NTHMP models in NTHMP (2012), taken from Tables 1–7 b in p. 38
Model error for case H = 0.019  

x = 9.95  x = 0.25  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
TsunamiHySEA  1  1  1  0  0.58  0.68 
Mean NTHMP (2012)  2  1  2  1  2  1 
3.1.3.4 Scalability
TsunamiHySEA has the option of solving dimensionless problems, and this is an option commonly used. When dimensionless problems are solved, it makes no sense to perform any test of scalability as the dimensionless problems to be solved for the different scaled problems will (if scaled to unity) always be the same.
3.2 Benchmark Problem #4: Simple Wave on a Simple Beach—Laboratory
This benchmark is the lab counterpart of BP1 (analytical benchmarking comparison). In this laboratory test, the 31.73mlong, 60.96cmdeep, and 39.97cmwide wave tank located at the California Institute of Technology, Pasadena was used with water of varying depths. The set of laboratory data obtained has been extensively used for many code validations. In this BP4, the datasets for the H/d = 0.0185 nonbreaking and H/d = 0.30 breaking solitary waves are used for code validation. The model has been first run in nonlinear, nondispersive mode. Then a dispersive version of TsunamiHySEA has also been used to assess the influence of dispersive terms in both, nonbreaking and breaking cases, and in both wave shape evolution and maximum runup estimation.
3.2.1 Problem Setup

Friction: Manning coefficient was set to 0.03 for the nondispersive model and slightly adjusted for the dispersive model (0.036 for the H/d = 0.30 and 0.032 for H/d = 0.0185).

Parameters: d = 1, g = 1, and H = 0.0185 for the nonbreaking Case And H = 0.30 for the breaking case.

Computational domain: the computational domain in x spanned from x = −10 to x = 70.

Boundary conditions: a nonreflective boundary condition at the right side of the computational domain is imposed.

Initial condition: the prescribed soliton at time t = 0 with the proposed initial velocity. These are the same conditions as for previous benchmark problem.

Grid resolution: the numerical results presented are for a computational mesh composed of 1600 cells, i.e., Δx = 0.05 = d/20.

Time stepping: variable time stepping based on a CFL condition.

CFL: CFL number is set to 0.9

Versions of the code: TsunamiHySEA thirdorder (with Marquina’s reconstruction) and secondorder (with MUSCL reconstruction) nondispersive models and secondorder (with MUSCL reconstruction) dispersive model have been benchmarked using this particular problem. Both nondispersive models give nearly identical results. In this case, dispersion plays an important role.
3.2.2 Tasks to be Performed
 1.
Compare numerically calculated surface profiles at t/T = 30:10:70 for the nonbreaking case H/d = 0.0185 with the lab data (Case A).
 2.
Compare numerically calculated surface profiles at t/T = 15:5:30 for the breaking case H/d = 0.3 with the lab data (Case C).
 3.
Numerically compute maximum runup (Case A and C).
 4.
Numerically compute maximum runup R/d vs. H/d.
3.2.3 Numerical Results
In this section, we present the numerical results obtained using TsunamiHySEA for BP4 according to the tasks to be performed as given in the benchmark description.
3.2.3.1 Water Level at Times t = 30, 40, 50, 60, and 70 (d/g)^{1/2} for Case A (H/d = 0.0185)
TsunamiHySEA model surface profile errors with respect to the lab experiment for Case A, H = 0.0185 at times t = 30:10:70 (d/g)^{1/2}. The values for NTHMP models are taken or computed from data in Table 1–8 a in p. 41 in NTHMP (2012)
Model error for CASE H = 0.0185  

t = 30  t = 40  t = 50  t = 60  t = 70  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
NDH  10.35  5.83  6.72  2.27  3.52  9.88  3.13  2.69  9.15  8.44  6.57  5.82 
NDN  11  6  9  3  6  13  4  1  33  15  10  8 
DH  6.69  3.92  5.35  1.19  4.6  5.12  3.24  1.73  8.63  3.59  5.7  2.1 
DN  11  3  8  2  4  3  5  4  12  6  8  3.5 
AN  11  4  8  3  5  7  5  3  16  9  9  5 
3.2.3.2 Water Level at Times t = 10, 15, 20, 25, and 30 (d/g)^{1/2} for Case C (H/d = 0.3)
TsunamiHySEA model surface profile errors with respect to the lab experiment for Case C, H = 0.30 at times t = 15:5:30 (d/g)^{1/2}. Nondispersive, dispersive model results and the mean of the four models with dispersion in NTHMP (2012) that presented results for this test are collected in this table. The values for NTHMP models are taken from data in Tables 1–8 b in p. 41 in NTHMP (2012)
Model error for CASE H = 0.30  

t = 15  t = 20  t = 25  t = 30  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
Nondispersive  22.5  17.33  17.42  52.34  5.17  10.07  2.32  3.09  11.85  20.70 
Dispersive  2.25  0.25  3.63  3.84  5.69  11.97  2.28  0.70  3.46  4.18 
Mean NTHMP  7  6  9  11  6  10  4  6  6.5  8 
3.2.3.3 Maximum runup (Case A and C)
3.2.3.4 Maximum Runup R/d vs. H/d
It can be observed that both nondispersive and dispersive models perform well in the case of the nonbreaking wave. Nevertheless, this same behavior does not occur for the breaking wave case. It can be seen, from Fig. 9, that the nondispersive model is not able to capture the time evolution of the wave in this particular case, tending to produce a shock wave that travels faster than the actual dispersive wave. Nevertheless, we observe that when the propagation phase ends and the inundation step takes place, the nondispersive model closely reproduces the observed new wave. Finally, regardless of whether we are simulating the breaking or nonbreaking wave, if we simply look at the runup time evolution we observed that both nondispersive and dispersive models produce quite close simulated time series (Fig. 11).
3.3 Benchmark Problem #6: Solitary Wave on a Conical Island—Laboratory
Three cases (A, B, and C) were performed corresponding to three wavemaker paddle trajectories.

CASE B: water depth, d = 32.0 cm, target H = 0.10, measured H = 0.096 (this case was formerly optional).

CASE C: water depth, d = 32.0 cm, target H = 0.20, measured H = 0.181.
To perform the tasks described below in Sect. 3.3.2.

CASE A: water depth, d = 32.0 cm, target H = 0.05, measured H = 0.045
In any case, we will include the three cases for all the tasks but for the splitting–colliding item.
3.3.1 Problem Setup

Friction: Manning coefficient is set to 0.015 for the nondispersive model and to 0.02 for the dispersive model.

Computational domain: [−5, 23] × [0, 28] in meters.

Boundary conditions: open boundary conditions.

Initial condition: the prescribed soliton centered at x = 0 with the proposed correction for the initial velocity (same expression as in BP1 and BP4, but extended to two dimensions, with wave elevation constant and zero velocity in the ydirection).

Grid resolution: for the nondispersive model a spatial grid resolution of 5 cm is used for Case A and a 2cm resolution grid for Cases B and C. Dispersive model uses a 2cm resolution for the three cases.

Time stepping: variable time stepping based on a CFL condition.

CFL: 0.9

Versions of the code: TsunamiHySEA second order with MUSCL reconstruction (nondispersive) and secondorder dispersive with MUSCL reconstruction codes have been used.
3.3.2 Tasks to be Performed
 1.
Demonstrate that two wavefronts split in front of the island and collide behind it;
 2.Compare computed water level with laboratory data at gauges 9, 16, and 22 (see Fig. 13 for graphical location and Table 5 for actual coordinates);Table 5
Laboratory gage positions. See Fig. 13 for graphical location
Gage ID
X (m)
Y (m)
Z (cm)
Comment
6
9.36
13.80
31.7
270° Transect
9
10.36
13.80
8.2
270° Transect
16
12.96
11.22
7.9
180° Transect
22
15.56
13.80
8.3
90° Transect
 3.
Compare computed island runup with laboratory gage data.
3.3.3 Numerical Results
Note that as we used the MATLAB scripts and data provided by J. Horrillo (Texas A&M University), we decided to perform numerical experiments for all the three cases A, B, and C, and also to present water level at gauge 6, although not included as mandatory requirements. For this benchmark, we have used TsunamiHySEA nondispersive and dispersive codes and have compared shape wave evolution and final maximum runup.
3.3.3.1 Wave Splitting and Colliding
3.3.3.2 Water Level at Gauges
Sea level time series TsunamiHySEA model error with respect to laboratory experiment data for Case A (H = 0.045). Comparison with the mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among nondispersive (Alaska, Geoclaw, and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for NTHMP models are taken from data in Tables 1–9 a in p. 46
Sea level model error for CASE A (H = 0.045)  

Gauge # 6  Gauge # 9  Gauge # 16  Gauge # 22  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
TsunamiHySEA  10  3  9  5  9  5  8  10  9  5.6 
Mean NTHMPND  6  9  7  14  10  10  8  25  8  15 
TsunamiHySEAD  9  2  9  3  8  2  8  9  8.3  3.9 
Mean NTHMPD  8  7  8  9  9  12  8  12  8  10 
Mean All NTHMP  7  8  8  10  9  12  8  18  8  12 
Sea level time series TsunamiHySEA model error with respect to laboratory experiment data for Case B (H = 0.096). Comparison with the mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among nondispersive (Alaska, Geoclaw, and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for NTHMP models are taken from data in Tables 1–9 b in p. 46
Sea level model error for CASE B (H = 0.096)  

Gauge # 6  Gauge # 9  Gauge # 16  Gauge # 22  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
TsunamiHySEA  9  1  8  4  10  1  9  10  9  4 
Mean NTHMPND  8  6  9  7  7  7  9  40  8  15 
TsunamiHySEAD  8  3  7  5  9  1  6  0  7.6  2.4 
Mean NTHMPD  7  6  8  10  6  7  10  20  8  11 
Mean All NTHMP  8  6  8  9  7  7  9  27  8  12 
Sea level time series TsunamiHySEA model error with respect to laboratory experiment data for Case C (H = 0.181). Comparison with the mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among nondispersive (Alaska, Geoclaw, and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for NTHMP models are taken from data in Tables 1–9 c in p. 46
Sea level model error for CASE C (H = 0.181)  

Gauge # 6  Gauge # 9  Gauge # 16  Gauge # 22  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
TsunamiHySEA  8  7  11  2  10  12  7  10  9  8 
Mean NTHMPND  10  6  11  9  9  3  8  18  9  9 
TsunamiHySEAD  7  0  10  7  8  6  6  2  7.9  3.9 
Mean NTHMPD  7  3  11  16  7  4  9  12  8  9 
Mean All NTHMP  8  5  11  13  8  3  8  15  9  9 
It can be observed that as we increase the value of H moving from Case A to B and Case C, the mismatch between the simulated wave and the measured one increases for the nondispersive model. The differences mostly increase in the leading wave. On the other hand, the dispersive model performs equally well in all the three cases.
3.3.3.3 Runup Around the Island
Runup TsunamiHySEA model error with respect to laboratory experiment data for all Cases A, B, and C. Comparison with the mean value obtained for the eight models performing this benchmark in NTHMP (2012) separated among nondispersive (Alaska, Geoclaw, and MOST) and dispersive models (ATFM, BOSZ, FUNWAVE, NEOWAVE, and SELFE), for a more precise comparison. The values for NTHMP models are taken from data in Tables 1–10 in p. 47
Runup model error  

CASE A (H = 0.045)  CASE B (H = 0.096)  CASE C (H = 0.181)  Mean  
RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  RMS (%)  MAX (%)  
TsunamiHySEA  7  0  19  1  5  0  10  0 
Mean NTHMPND  18  12  21  2  12  5  17  7 
TsunamiHySEAD  8  0  4  4  6  5  6  3 
Mean NTHMPD  17  4  16  7  10  5  15  5 
Mean All NTHMP  18  7  18  5  11  5  16  5 
In this benchmark, the observed behavior of the simulated maximum runup for nondispersive and dispersive models through the three cases considered is not so easily explained. Now for cases A and B, in the extremes, both models perform similarly well. In Case B, the nondispersive model performs clearly worse, while the dispersive model performs equally well.
3.4 Benchmark Problem #7: The Tsunami Runup onto a Complex ThreeDimensional Model of the Monai Valley Beach—Laboratory
3.4.1 Problem Setup

Friction: Manning coefficient is set to 0.03

Computational domain: [0, 5.488] × [0, 3.402] (units in meters).
 Boundary conditions: the given initial wave (Fig. 25) was used to specify the boundary condition at the left boundary up to time t = 22.5 s; after time t = 22.5 s, nonreflective boundary conditions. Solid wall boundary conditions were used at the top and bottom boundaries.

Initial condition: water at rest.
 Grid resolution: a 393 × 244size mesh was used, with the same resolution (0.014 m) as the bathymetry. Table 10 collects grid information.Table 10
Mesh information showing grid resolution, number of cells and computing time needed for a 200s simulation
Grid resolution
Δx = Δy (m)
# of volumes
Comput. time [s(min)]
393 × 244
0.014
95,892
91.54618
(1.52)

Time stepping: variable time stepping based on a CFL condition.

CFL: 0.9

Versions of the code: TsunamiHySEA second order with MUSCL reconstruction and WAF models used for this benchmark. Secondorder model results are presented.
3.4.2 Tasks to be Performed
 1.
Model the propagation of the incident and reflective wave according to the benchmarkspecified boundary condition.
 2.
Compare the numerical and laboratorymeasured water level dynamics at gauges 5, 7, and 9 (in Fig. 24).
 3.
Show snapshots of the numerically computed water level at time synchronous with those of the video frames; it is recommended that each modeler finds times of the snapshots that best fit the data.
 4.
Compute maximum runup in the narrow valley.
3.4.3 Numerical Result
In this section, we present the numerical results for BP7 as simulated by TsunamiHySEA according to the tasks to be performed as given in the benchmark description.
3.4.3.1 Gauge Comparison
3.4.3.2 Frame Comparisons
3.4.3.3 Runup in the Valley
3.5 Benchmark Problem #9: Okushiri Island Tsunami—Field
The goal of this benchmark problem is to compare computed model results with field measurements gathered after the 12 July 1993 Hokkaido NanseiOki tsunami (also commonly referred to as the Okushiri tsunami).
3.5.1 Problem setup

Friction: Manning coefficient 0.03.

Boundary conditions: nonreflective boundary conditions at open sea, at coastal areas inundation is computed.
 Computational domain: a nested mesh technique is used with four levels (i.e., the global mesh with three levels of refinement, see Figs. 29 and 30).
 Global mesh coverage in lon/lat [138.504, 140.552] × [41.5017, 43.2984].

Number of cells: 1152 × 1011 = 1,164,672.

Resolution: 6.4 arcsec (≈192 m).

 Level 1. Spatial coverage [139.39, 139.664] × [41.9963, 42.2702].

Refinement ratio: 4.

Number of cells: 616 × 616 = 379,456.

Resolution: 1.6 arcsec (≈40 m).

 Level 2. Refinement ratio: 4. Resolution: 0.4 arcsec (≈12 m).
 Submesh 1: large area around Monai.

Spatial coverage [139.434, 139,499] × [42.0315, 42.0724].

Number of cells: 584 × 368 = 214,912.

 Submesh 2: Aonae cape and Hamatsumae region.

Spatial coverage [139.411, 139.433] × [42.0782, 42.1455].

Number of cells: 196 × 604 = 118,384.


 Level 3 (Monai region). Spatial coverage [139.414, 139.426] × [42.0947, 42.1033].

Refinement ratio: 16.

Number of cells: 1744 × 1248 = 2,216,448.

Resolution: 0.025 arcsec (≈0.75 m).



Initial condition: generated by DCRC (Disaster Control Research Center), Japan. Hipocenter depth 37 km at 139.32°E and 42.76°N, M _{w} 7.8 (Takahashi 1996) (Fig. 29, source model DCRC 17a).

Topobatymetric data: Kansai University.

CFL: 0.9.

Version of the code: TsunamiHySEA WAF.
3.5.2 Tasks to be Performed
 1.
Compute runup around Aonae.
 2.
Compute arrival of the first wave to Aonae.
 3.
Show two waves at Aonae approximately 10 min apart; the first wave came from the west, the second wave came from the east.
 4.
Compute water level at Iwanai and Esashi tide gauges.
 5.
Maximum modeled runup distribution around Okushiri Island.
 6.
Modeled runup height at Hamatsumae.
 7.
Modeled runup height at a valley north of Monai.
3.5.3 Numerical Results
In this section, the numerical results obtained with TsunamiHySEA for BP9 are presented.
3.5.3.1 Runup Around Aonae
3.5.3.2 First Wave to Aoane
3.5.3.3 Waves Arriving to Aonae
3.5.3.4 Tide gauges at Iwanai and Esashi
3.5.3.5 Maximum Runup Around Okushiri
TsunamiHySEA model relative error with respect to field measurement data for runup around Okushiri Island. Comparison with average error values for models in NTHMP (2102). #OBS gives the number of observations used to compute the error bars in Fig. 37
Region  Longitude  Latitude  # OBS  HySEA (%)  Mean NTHMP (%) 

1  139.4292117  42.18818149  3  25  5 
2  139.4111857  42.16276287  2  22  8 
3  139.4182612  42.13740439  1  66  27 
4  139.4280358  42.09301238  1  2  7 
5  139.4262450  42.11655479  1  5  6 
6  139.4237147  42.10041415  7  2  6 
7  139.4289018  42.07663658  1  30  15 
8  139.4278534  42.06546152  2  5  10 
9  139.4515399  42.04469655  3^{a}  0  0 
10  139.4565284  42.05169226  5^{a}  0  8 
11  139.4720138  42.05808988  4  0  2 
12  139.5150461  42.21524909  2  0  10 
13  139.5545494  42.22698164  6–8  19  14 
14  139.4934307  42.06450128  3  32  74 
15  139.5474599  42.18744879  1  6  14 
16  139.5258982  42.17101221  2  0  11 
17  139.5625242  42.21198369  1  9  15 
18  139.5190997  42.11305805  3  0  34 
19  139.5210766  42.15137635  2  1  19 
Mean  12  15 
3.5.3.6 Runup Height at Hamatsumae
3.5.3.7 Runup Height at a Valley North of Monai
4 Conclusions
The TsunamiHySEA numerical model is validated and verified using NOAA standards and criteria for inundation. The numerical solutions are tested against analytical predictions (BP1, solitary wave on a simple beach), laboratory measurements (BP4, solitary wave on a simple beach; BP6, solitary wave on a conical island; and BP7, runup on Monai Valley beach), and against field observations (BP9, Okushiri island tsunami). In the numerical experiments modeling the propagation and runup of a solitary wave on a canonical beach, numerical results are clearly below the established errors by the NTHMP in their 2011 report. For BP1, the mean errors measured are below 1% in all cases. In the case of BP4, several conclusions can be extracted. For the nonbreaking case with H = 0.0185 the nondispersive model produces accurate wave forms with NRMSD errors, in most cases, very close to the dispersive model results. For the breaking wave case with H = 0.30 it can be observed that the shape of the (dispersive) wave cannot be well captured by the nondispersive model, producing large NRMSD errors at the times when the NLSW model tends to produce a shock. Nevertheless, the agreement is still high for times when nonsteep profiles are present. Despite this (a dispersive model is absolutely necessary if we want to accurately reproduce the time evolution of the wave in the breaking case) we have observed that measured runup is accurately reproduced by both models in the two studied cases. On the other hand, the dispersive version of TsunamiHySEA produces very good results in both the breaking and nonbreaking cases. For BP6, dealing with the impact of a solitary wave on a conical island, again nondispersive and dispersive TsunamiHySEA models have been used. Wave splitting and colliding are clearly observed. Numerical results are very similar for Case A (A/h = 0.045) and Case B (A/h = 0.096) for wave shape. Larger differences are evident in Case C (A/h = 0.181), where dispersive model performs better for wave shape, but not for the computed runup. It is noteworthy that the computed maximum runups for Cases A and C are very close for both models but they clearly differ for Case B. TsunamiHySEA model figures have been compared with figures in NTHMP (2012), performing in general better than the mean when comparing by class of model (dispersive and nondispersive). BP7, the laboratory experiment dealing with the tsunami runup onto a complex 3D model of the Monai Valley beach, was studied in detail in (Gallardo et al. 2007). A mean value of 7.66% for the NRMSD is obtained for all the three gauges for the times series simulating the first 30 s. The snapshots of the simulation agree well with the experimental frames and, finally, a maximum simulated runup height of 0.0891 is obtained compared with the 0.08958 experimentally measured. Comparison of BP9 with Okushiri island tsunami observed data is performed using nested meshes with two level 2 meshes located one in the South of the island, covering Aoane and Hamatsumae areas and the second one to the West containing Monai area. Finally, one level 3 refined mesh is located covering the Monai area. Computed runup and arrival times are in good agreement with observations. Water level time series at Iwanai and Esashi tide gauges show large NRMSD and large errors in the maximum amplitude (36 and 41% for ERR) but analogous to the mean of the models in NTHMP (2012) (36 and 43% for ERR). For the maximum runup at 19 regions around Okushiri Island a mean error of 15% is obtained, the same as the mean of models in NTHMP (2012), with 10 regions with errors below 10%. Regions located in areas with refined meshes perform much better than regions located in coarse mesh areas.
Notes
Acknowledgements
This research has been partially supported by the Spanish Government Research project SIMURISK (MTM201570490C21R), the Junta de Andalucía research project TESELA (P11RNM7069), and Universidad de Málaga, Campus de Excelencia Internacional Andalucía Tech. The GPU and multiGPU computations were performed at the Unit of Numerical Methods (UNM) of the Research Support Central Services (SCAI) of the University of Malaga.
References
 Apotsos, A., Buckley, M., Gelfenbaum, G., Jafe, B., & Vatvani, D. (2011). Pure and Applied Geophysics, 168(11), 2097–2119. doi: 10.1007/s0002401102915.CrossRefGoogle Scholar
 Briggs, M., Synolakis, C., Harkins, G., & Green, D. (1995). Laboratory experiments of tsunami runup in a circular island. Pure and Applied Geophysics, 144, 569–593.CrossRefGoogle Scholar
 Castro, M.J., de la Asunción, M., Macías, J., Parés, C., FernándezNieto, E.D., GonzálezVida, J.M., Morales, T. (2012). IFCP Riemann solver: Application to tsunami modelling using GPUs. In E. Vázquez, A. Hidalgo, P. García, L. Cea eds. CRC Press. Chapter 5, 237–244.Google Scholar
 Castro, M. J., & FernándezNieto, E. D. (2012). A class of computationally fast first order finite volume solvers: PVM methods. SIAM Journal on Scientific Computing, 34, A2173–A2196.CrossRefGoogle Scholar
 Castro, M. J., FernándezNieto, E. D., Ferreiro, A. M., GarcíaRodríguez, J. A., & Parés, C. (2009). High order extensions of Roe schemes for twodimensional nonconservative hyperbolic systems. Journal of Scientific Computing, 39(1), 67–114.CrossRefGoogle Scholar
 Castro, M. J., Ferreiro, A., García, J. A., González, J. M., Macías, J., Parés, C., et al. (2005). On the numerical treatment of wet/dry fronts in shallow flows: applications to onelayer and twolayer systems. Mathematical and Computer Modelling, 42(3–4), 419–439.CrossRefGoogle Scholar
 Castro, M. J., González, J. M., & Parés, C. (2006). Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme. Mathematical Models and Methods in Applied Sciences, 16(6), 897–931.CrossRefGoogle Scholar
 Castro, M. J., Ortega, S., Asunción, M., Mantas, J. M., & Gallardo, J. M. (2011). GPU computing for shallow water flow simulation based on finite volume schemes. Comptes Rendus Mécanique, 339, 165–184.CrossRefGoogle Scholar
 de la Asunción, M., Castro, M. J., FernándezNieto, E. D., Mantas, J. M., Ortega, S., & GonzálezVida, J. M. (2013). Efficient GPU implementation of a two waves TVDWAF method for the twodimensional one layer shallow water system on structured meshes. Computers & Fluids, 80, 441–452.CrossRefGoogle Scholar
 de la Asunción, M., Mantas, J. M., & Castro, M. J. (2011). Simulation of onelayer shallow water systems on multicore and CUDA architectures. The Journal of Supercomputing, 58, 206–214.CrossRefGoogle Scholar
 Gallardo, J. M., Ortega, S., de la Asunción, M., & Mantas, J. M. (2011). Twodimensional compact thirdorder polynomial reconstructions. Solving nonconservative hyperbolic systems using GPUs. Journal of Scientific Computing, 48, 141–163.CrossRefGoogle Scholar
 Gallardo, J. M., Parés, C., & Castro, M. J. (2007). On a wellbalanced highorder finite volume scheme for shallow water equations with topography and dry areas. Journal of Computational Physics, 227, 574–601.CrossRefGoogle Scholar
 Gottlieb, S., & Shu, C. W. (1998). Total variation diminishing RungeKutta schemes. Mathematics of Computation, 67, 73–85.CrossRefGoogle Scholar
 Horrillo, J., Grilli, S. T., Nicolsky, D., Roeber, V., & Zhang, J. (2015). Performance benchmarking Tsunami models for nthmp’s inundation mapping activities. Pure and Applied Geophysics, 172(3), 869–884. doi: 10.1007/s000240140891y.CrossRefGoogle Scholar
 Kato, H., & Tsuji, Y. (1994). Estimation of fault parameters of the 1993 HokkaidoNanseiOki earthquake and tsunami characteristics. Bulletin of the Earthquake Research Institute, 69, 39–66.Google Scholar
 Liu, P.L.F., Yeh, H., & Synolakis, C. (2008). Advanced numerical models for simulating Tsunami waves and runup. Advances in coastal and ocean engineering (vol. 10). New Jersey: World Scientific.Google Scholar
 Macías, J., Castro, M.J., GonzálezVida, J.M., de la Asunción, M., and Ortega, S. (2014). HySEA: An operational GPUbased model for Tsunami Early Warning Systems. EGU 2014.Google Scholar
 Macías, J., Castro, M.J., GonzálezVida, and J.M., Ortega, S. (2013). Nonlinear Shallow Water Models for coastal runup simulations. EGU 2013.Google Scholar
 Macías, J., Castro, M.J., GonzálezVida, J.M., Ortega, S., and de la Asunción, M. (2013). HySEA tsunami GPUbased model. Application to FTRT simulations. International Tsunami Symposium (ITS2013). Göcek (Turkey). 25–28 September 2013.Google Scholar
 Macías, J., Castro, M.J., Ortega, S., Escalante, C., and GonzálezVida, J.M. (2016). TsunamiHySEA Benchmark results. In NTHMP report for the MMS Benchmarking Workshop: Tsunami Currents.Google Scholar
 Macías, J., Mercado, A., GonzálezVida, J.M., Ortega, S., and Castro, M.J. (2015). Comparison and numerical performance of TsunamiHySEA and MOST models for LANTEX 2013 scenario. Impact assessment on Puerto Rico coasts. Pure and Applied Geophysics. doi: 10.1007/s0002401613878.
 Marquina, A. (1994). Local piecewise hyperbolic reconstructions for nonlinear scalar conservation laws. SIAM Journal of Scientific Computing, 15, 892–915.CrossRefGoogle Scholar
 National Tsunami Hazard Mitigation Program (NTHMP). 2012. Proceedings and Results of the 2011 NTHMP Model Benchmarking Workshop. Boulder: U.S. Department of Commerce/NOAA/NTHMP; NOAA Special Report. p. 436.Google Scholar
 National Tsunami Hazard Mitigation Program (NTHMP). 2016. Report on the 2015 NTHMP Current Modeling Workshop. Portland, Oregon. p. 202.Google Scholar
 Nicolsky, D. J., Suleimani, E. N., & Hansen, R. A. (2011). Validation and verification of a numerical model for Tsunami propagation and runup. Pure and Applied Geophysics, 168(6), 1199–1222.CrossRefGoogle Scholar
 Parés, C. (2006). Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM Journal on Numerical Analysis, 44(1), 300–321.CrossRefGoogle Scholar
 Roeber, V., Cheung, K. F., & Kobayashi, M. H. (2010). Shockcapturing Boussinesqtype model for nearshore wave processes. Coastal Engineering, 57, 407–423.CrossRefGoogle Scholar
 Synolakis, C. E. (1987). The runup of long waves. Journal of Fluid Mechanics, 185, 523–545.CrossRefGoogle Scholar
 Synolakis, C. E., Bernard, E. N., Titov, V. V., Kânoğlu, U., & González, F. I. (2008). Validation and verification of tsunami numerical models. Pure and Applied Geophysics, 165(11–12), 2197–2228.CrossRefGoogle Scholar
 Takahashi, T., 1996. Benchmark Problem 4. The 1993 Okushiri tsunami. Data, Conditions and Phenomena. In: H. Yeh, P. Liu, C. Synolakis (eds.): Long wave runup models. Singapore: World Scientific Publishing Co. Pte. Ltd., pp. 384–403.Google Scholar
 Tolkova, E. (2014). Landwater boundary treatment for a tsunami model with dimensional splitting. Pure and Applied Geophysics, 171(9), 2289–2314.CrossRefGoogle Scholar
 van Leer, B. (1979). Towards the Ultimate Conservative Difference Scheme, V. A Second Order Sequel to Godunov’s Method. Journal of Computational Physics, 32, 101–136.CrossRefGoogle Scholar
 Yamazaki, Y., Kowalik, Z., & Cheung, K. F. (2009). Depthintegrated, nonhydrostatic model for wave breaking and runup. International Journal for Numerical Methods in Fluids, 61(5), 473–497.CrossRefGoogle Scholar
 Yeh, H., Liu, P., Synolakis, C., editors (1996). Benchmark problem 4. The 1993 Okushiri Data, Conditions and Phenomena. Singapore: World Scientific Publishing Co. Pte. Ltd.Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.