Horizontal analytical solution to the Boussinesq equations

[jgalazm]

From @jgalazm on October 6, 2017 20:24

I could use, for instance, an analytical solution provided for the Boussinesq equations in 1D, along with the Imamura relation that makes the truncation error of the shallow water equations produce the same dispersion (up to O(dx^3,dtdx²,dxdt²,dt^3) of the Boussinesq equations.

One 2D simple solution, for uniform bottom is: adfasdf

Yoon, S. B. (2002). Propagation of distant tsunamis over slowly varying topography. Journal of Geophysical Research: Oceans, 107(C10).

is given by:

• Initial condition:

• Parameters for discretization: Two depths: h= 600m, 2100m Domain: r \in [0, 500km]

Copied from original issue: Inria-Chile/tsunami-lab#97

[jgalazm]

I'm restaring this one.

The initial condtion, analytical solution and dimensionless coordinates:

The idea is to compare with Y. Cho (195) "Numerical simulations of tsunami propagation and run-up"

Numerical setup

a = 1 h = 0.2 dx = 0.5 dt = 0.3/sqrt(9.81*0.2)

this way Imamura's condition is satisfied. dx² = 4h² + dt²

The first validation was done in this prototype notebook:

https://github.com/Inria-Chile/tsunami-lab/blob/develop/prototypes/comcot_linear_vectorized.ipynb

And the idea is to compare the result with this figure (3.3 on Cho's dissertation)

which corresponds to any radial profile at t'=40, i.e, t = t' a/sqrt(gh0) =~ 28.55s Probably at 28.6 given the timestep

However, now I can't set the timestep explicitly, I'm reopening issue #86

[jgalazm]

I should use a different snapshot of the analytical solution that uses a multiple of dt to show better accuracy

[jgalazm]

After the first implementation the results look like this, for dx = 2dx* and theta=0,45,90

To do now:

• [ ] Implement the one that requires dx=dx* but handles 2D grids
• [ ] Implement the one with the hidden grid
• [ ] Try different depths --> sometimes it crashes, even with small cfl
• [ ] Try scenario with smooth - variable depth
• [ ] Implement the one with h=ax+by+c