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Numerical Simulation of Benchmark Problem 2

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

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Numerical Simulation of Benchmark Problem 2

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  1. 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

  2. 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.

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

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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)

  9. 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.

  10. 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

  11. 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)

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

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

  14. Comparison between numerical simulations - Coarse grid vs. Finer grid

  15. Comparison with gage data – Coarse grid simulation

  16. Comparison with gage data – Finer grid simulation

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

  18. Animation – Runup Lab movie Entire region movie

  19. 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.

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