1. WORKSHOP ON LONG-WAVE RUNUP MODELS Khairil Irfan Sitanggang and Patrick Lynett Dept of Civil & Ocean Engineering, Texas A&M University

2. Governing Equations and Setupz x z ao ho h
H = h + z
e = a/ho
m = (ho/lo) lo

3. Governing Equations – Highly Nonlinear Boussinesq
Continuity Equation (Dimensionless)

4. Governing Equations – Highly Nonlinear Boussinesq
Momentum Equation (Dimensionless)

5. Numerical Algorithm Predictor - Corrector Scheme
3rd-order explicit Adams-Bashforth (Press et al., 1989) predictor step - (Dt)3
4th -order implicit Adams-Moulton corrector step - (Dt)4
Finite difference spatial derivatives to 4th-order accuracy - (Dx)4
Scheme developed to solve the Boussinesq equations in 2HD
Not a computationally efficient solver for the NLSW equations or 1HD setups

6. Practical Simulation
Bottom friction based on the quadratic friction law
Wave breaking with an eddy viscosity model
Comparison with data from Hansen & Svendsen (1979)
Runup/rundown uses the extrapolation moving shoreline procedure (Lynett et al, 2002)

7. Moving Boundary Algorithm
The underlying assumption is that very near the wet-dry boundary, the wave is linear in slope:
Linear extrapolation- imaginary points in dry region
The free surface and velocity are linearly extrapolated through the wet-dry boundary, creating “imaginary” points in the dry region.
Wet nodal points near the wet-dry boundary use the extrapolated points when calculating finite difference derivates (5-point centered differences)

8. Validation of Runup algorithm
Runup of solitary waves
Comparison with experimental data taken from Synolakis (1987)
Numerical simulation parameters:
Wave height / water depth = 0.04
Beach slope = 1:20
Comparison with experimental data:
Numerical results
Experimental data

9. Validation of Runup algorithm
Runup of solitary wave around a circular island
Experimental data taken from Liu et al. (1995)
Inundation comparisons
Black dots represent the maximum experimental runup, while the light red shows the inundated area

10. Benchmark #1 Initial free surface given Run with: NLSW Boussinesq
Will examine the dispersive effect & required grid resolution near shoreline for numerical convergence
Simulations have no bottom friction, wave breaking not included

11. Benchmark #1 Numerical convergence
Nearshore grid ~1.5 m is required for numerical convergence
Error much larger in velocity predictions

12. Benchmark #1 CPU requirements:
Numerical model uses a constant dx and dt
dt is chosen based on a Courant formulation, where the characteristic velocity is the long wave velocity in the deepest water depth in the domain.
With constant slope to offshore boundary (and very deep water) a small time step is required.
NLSW simulation:
dx=1.5 m, dt= s
nx~32,000, nt~170,000 (endx~50 km, endt~300s)
51 MB of RAM
40 hrs of CPU time on a 1.8 Ghz desktop!
~0.9 seconds per time step
Numerical implementation not developed for large 1HD problems or NLSW equations

13. Benchmark #1
Shoreline elevation and velocity time histories Boussinesq numerical vs NLSW numerical vs Analytical
NLSW numerical result matches solution very well
Boussinesq solution indicates that dispersive effects during shoaling may play a significant role

14. Benchmark #1
Snapshots of free surface and velocity at various times

15. Benchmark #1 Conclusions
Required small grid for convergent runup results leads to very large CPU times for a 1HD problem
Large grid error more significant for velocity comparisons
NLSW numerical results match well for both shoreline and spatial profiles
Boussinesq predicts higher maximum runup, but lower maximum speeds
Frequency dispersion may be important during shoaling, as the wave becomes very steep

16. Benchmark #2 Khairil Irfan Sitanggang, Patrick Lynett Dept of Civil & Ocean Engineering, Texas A&M University, U.S.A Alejandro Orfila IMEDEA (CSIC-UIB), Spain
CPU requirements:
Numerical model uses a constant dx and dt
To capture the oscillations inside the cove, a relatively small global grid size was used
Boussinesq simulation:
dx~0.012 m, dt~0.005 s
nx~450, ny~300, nt~4,500 (endt=25s)
200 MB of RAM
5.5 hrs of CPU time on a 1.8 Ghz desktop
~4.5 seconds per time step
Tsunami generation by internal source generator, using specified input wave time series

17. Benchmark #2 Tsunami Approach on Complex Bathymetry
Bottom friction included (quadratic friction law)
Friction coefficient =
Breaking model used
Wave begins to “break” when zt>0.65[g(h+z)]0.5
Energy dissipation (eddy viscosity model) is added at the breaking locations

18. Benchmark #2 Tsunami Approach on Complex Bathymetry
Animation from Boussinesq simulation (wide view)

19. Benchmark #2 Tsunami Approach on Complex Bathymetry
Animation from Boussinesq simulation (shoreline closeup)
Max predicted runup in cove ~ 6.5 cm.

20. Benchmark #2 Tsunami Approach on Complex Bathymetry
Comparison w/ video data
Angle of approach of the positive wave is different, leading to different runup patterns

21. Benchmark #2
Free surface time series comparisons

22. Benchmark #2 Conclusions
Model does an OK job at recreating the experiment
Primary differences due to different approach angle of the positive wave
Possible causes:
Breaking not predicted correctly
Bottom friction underestimated
Input wave not exact
No significant differences between NLSW and Boussinesq for this case

23. Benchmark #3 Khairil Irfan Sitanggang, Patrick Lynett Dept of Civil & Ocean Engineering, Texas A&M University, U.S.A
CPU requirements:
Case A:
dx=10 m, dt=0.16 s (small dt for deep water stability)
nx=470, nt=450
Boussinesq CPU time= 35 seconds on 1.8 Ghz desktop
Case B:
dx=0.25 m, dt=0.008 s
nx=400, nt=1400
Boussinesq CPU time= 90 seconds on 1.8 Ghz desktop
For reference, numerical integration of the analytical solution required:
nx=1300, nt=2000
CPU time = 6 hours on 1.8 Ghz desktop (lots of integration with Bessel & exponential functions)

24. Benchmark #3 Subaerial slide
Animation of Case A from Boussinesq Simulation
No bottom friction
Breaking model not implemented
Slide is shown as the yellow area
Slide very thin

25. Benchmark #3
CASE A comparisons (large tanb/m)

26. Benchmark #3 Subaerial slide
Animation of Case B from Boussinesq Simulation
No bottom friction
Breaking model not implemented

27. Benchmark #3
CASE B comparisons (small tanb/m)

28. Benchmark #3 Conclusions Case A:
Except for very early time, agreement excellent
Shoreline does not move much
NLSW and Boussinesq identical
Case B:
Agreement with analytical solution OK at early time, but differences grow quickly
The analytical solution, due to assumptions on the slide forcing, is not adequate for this case
Some difference between NLSW and Boussinesq at later times, but relatively minor
Indications that turbulence and/or dispersive effects become important at later times