1. Numerical Simulation of One-dimensional Wave Runup by CIP-like Moving Boundary Condition Koji FUJIMA National Defense Academy of Japan

2. Cubic-Interpolated Pseudo-particle [Yabe; 1988, 1991] Basic Idea (1) : pseudo-particle The Solution of can be approximated as Basic Idea (2) : cubic-polynomial interpolation in is interpolated by cubic-polynomial by using, where f’ is the derivative with respect to x.

3. CIP algorithm for Phase 1 = non-advection phase Phase 2 = advection phase Pseudo-particle Cubic-polynomial Interpolation

4. FEM + CIP (Ishikawa et al., 2003)

5. CIP method Very easy and applicable to solve the various hyperbolic equations (one-dimensional, multi- dimensional, one-variable, multi-variables,,,) All variables are defined usually at the same time step.

6. The present method The main part of simulation is same as the conventional method. (Staggered leapflog mesh) Moving boundary (wave front) is evaluated by CIP-like (pseudo-particle) manner. FEM+CIP seems too heavy for the practical tsunami simulation. The conventional tsunami simulation has sufficient cost-performance for many problems.

7. Moving Boundary in the Conventional method When, M=0 and the wave front does not move. When, M is computed and the wave front moves.

8. Numerical scheme of the present model: ( and are assumed to be determined) Equation of continuity : extrapolated by u behind the wave front

9. Equation of motion The advection and pressure gradient terms are evaluated by the same manner as the conventional tsunami simulation. When M behind the front is computed, pressure gradient is estimated as the right figure.

10. Extrapolation of Cubic-extrapolation Linear-extrapolation Same as closest grid The simplest method is adopted.

11. Wave profile and velocity distribution at t=160s

12. Wave profile and velocity distribution at t=175s

13. Wave profile and velocity distribution at t=220s

14. Trajectory of wave front location and front velocity x(numerical), x(theoretical) u(numerical), u(theoretical)

15. Trajectory of wave front location and front velocity x(numerical-conventional model), x(theoretical) u(numerical-conventional model), u(theoretical)

16. Summary The only difference of the conventional model and the present model is the treatment of the moving boundary. The computation cost is almost similar in both models, although those results are quite different. The conventional model underestimates the wave runup and rundown. The present model tend to underestimate the wave rundown and overestimate the wave runup.