Numerical Simulation of Benchmark Problem 2 Xiaoming Wang Philip L.-F. Liu and Alejandro Orfila Dept. of Civil Engineering, Cornell University The 3rd Intl. Workshop on long wave propagations Catalina Island, June 17-19, 2004

Numerical Method - SWE Program: COMCOT COMCOT is a tsunami simulation program. It adopts finite difference scheme to solve the depth-averaged Shallow Water Equations. Multiple nested grids can be employed simultaneously to save CPU time as well as obtain enough resolution at target region.

Governing Equations COMCOT solves the depth-averaged Shallow Water Equations. Linear equations:

Governing Equations Nonlinear equations: For both linear and nonlinear equations, P=Hux and Q=Huy. H=+h.  is the free surface displacement; h is the still water depth; and H is the total water depth.

Governing Equations The bottom friction is expressed as which come from Manning’s formula. n is roughness coefficient. In this simulation, n takes 0.01.

Finite difference scheme COMCOT uses an explicit leap-frog scheme : The free surface elevation is evaluated at the center of a grid cell on the (n+1/2)-th time step; The volume flux components, P and Q, are evaluated at the center of four sides of the grid cell on the n-th time step. The differencing schemes are shown in the right figure.

Moving boundary scheme The following figure shows a 1-D example of the moving boundary scheme used in COMCOT. Moving boundary applies When Hi>0 and Hi+1<=0: 1. If hi+1 + i<0, shoreline at i+1/2 and Pi+1/2=0. Total water depth is 0 at cell i+1. 2. If hi+1 + i>0, shoreline moves between i+1 and i+2. Pi+1/2 may have a none-zero value. Total water depth is H=hi+1 + i at cell i+1. Where, h - takes positive values for water regions (wet cells) and negative values for land region (dry cells). Initially =-h at dry cell at t=0.

Computational domain Left boundary : input wave boundary, Shut down at t = 30s Right boundary : solid wall Top boundary : solid wall Bottom boundary: solid wall Bilinear interpolation is used to get a higher resolution (for dx<0.01435m)

Initial water surface profile The given initial surface profile has been modified to smooth the sudden change at the tail. Linear interpolation is adopted to get a higher resolution.

Numerical Simulations Coarse grid simulations: 1. Using linear Shallow Water Equations (without bottom friction) 2. Using nonlinear Shallow Water Equations (w/o bottom friction) configurations for both cases: dx=0.01435m, dimension: 393*244 dt=0.001s, Courant No. = 0.08 Roughness coeff. = 0.01 (Manning’s formula, if using bottom friction) Finer grid simulation Using nonlinear Shallow Water Equations. Configuration: dx=0.005m, dimension: 1098*681 dt=0.0002s, Courant No. = 0.05 without bottom friction

Numerical characteristics Platform - IBM compatible PC OS : Windows 2000 Professional CPU: AMD Athlon XP 2600+ (2.13 GHz) RAM: 1.0 GB DDR400 Dual Channel CPU Time 1. For coarse grid simulation (nonlinear): 0.074s per step (total steps: 150000) Total CPU time: 3.08hrs (for 150s physical simulation) 2. 1.193s per step For finer grid simulation (nonlinear eq.): 1.193s per step (total steps: 200000) Total CPU time: 2days+18hrs (for 40s physical simulation)

Comparison between numerical simulations - Linear LSWE vs Comparison between numerical simulations - Linear LSWE vs. Nonlinear NLSWE (without bottom friction)

Comparison between numerical simulations – bottom friction vs Comparison between numerical simulations – bottom friction vs. no bottom friction (nonlinear eq.)

Comparison between numerical simulations - Coarse grid vs. Finer grid

Comparison with gage data – Coarse grid simulation

Comparison with gage data – Finer grid simulation

Runup – Coarse grid, nonlinear SWE Maximum runup - with bottom friction : 9.1 cm - without bottom friction: 10.2 cm

Animation – Runup Lab movie Entire region movie

Conclusions The numerical results show that the problem can be well simulated as a first attempt with the linear system without including bottom friction. An agreement between grid size and time consumptions has to be considered since the reduction of the grid does not lead to a much accurate results. COMCOT results match the records very good for both the arrival time and amplitude of leading waves. In the near shore region, the waves becomes very nonlinear and will break. COMCOT is no longer capable to deal with it. The wave amplitude may be exaggerated.