Numerical Simulation of One-dimensional Wave Runup by CIP-like Moving Boundary Condition

Koji FUJIMA

National Defense Academy of Japan

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.

CIP algorithm for

• Phase 1 = non-advection phase

• Phase 2 = advection phase

• Pseudo-particle• Cubic-polynomial Interpolation

FEM + CIP (Ishikawa et al., 2003)

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.

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.

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 .

Numerical scheme of the present model:( and are assumed to be determined)

• Equation of continuity

: extrapolated by u behind the wave front

• 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.

Extrapolation of

• Cubic-extrapolation

• Linear-extrapolation

• Same as closest grid

The simplest method is adopted.

Wave profile and velocity distribution at t=160s

Wave profile and velocity distribution at t=175s

Wave profile and velocity distribution at t=220s

Trajectory of wave front location and front velocityx(numerical) , x(theoretical) u(numerical) , u(theoretical)

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

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.