# Boussinesq solvers in One Space Dimension¶

Warning

Not yet incorporated in clawpack master branch or releases.

As of Version 5.10.0 (?), the geoclaw repository contains some code for solving problems in one space dimension, as described more generally in GeoClaw in One Space Dimension. This code also supports two different sets of dispersive Boussinesq equations that have been used in the literature to better model wave propagation in situations where the wavelength is not sufficiently long relative to the fluid depth for the shallow water equation approximation to be accurate.

These Boussinesq equations are still depth-averaged equation with the same conserved quantities (fluid depth h and momentum hu in 1d), but the equations contain higher order derivative terms and so they are no longer hyperbolic. The equations implemented include third-order derivatives with respect to txx. However, the implementations proceed by alternating steps with the shallow water equations and the solution of elliptic equations that involve only second-order derivatives in xx.

## The SGN equations¶

The recommended set of equations to use are a modification of the Serre-Green-Naghdi (SGN) equations with the addition of a parameter alpha suggested by Bonneton et al. Both the 1d and 2d versions of these equations and their GeoClaw implementation are discussed in [BergerLeVeque2023]. Setting alpha = 1 gives the original SGN equations, but alpha = 1.153 is recommended since it gives a better approximation to the linear dispersion relation of the Airy solution to the full 3d problem.

## The Madsen-Sorensen (MS) equations¶

These equations also give a good approximation to the linear dispersion relation of the Airy solution when the parameter beta = 1/15 is used. These equations were used in an earlier GeoClaw implementation known as BoussClaw. These have been reimplemented in GeoClaw more recently, including a 2d implementation that allows the use of AMR. However, extensive tests with these equations have revealed some stability issues, particularly in the case of AMR, which have also been reported by others. The 1d MS implementation is included in GeoClaw but it is generally not recommended except for those interested in comparing different formulations and perhaps further investigating the stability issues.

## Using the 1d Boussinesq code¶

As in the 1d shallow water implementation, general mapped grids can be used, but AMR is not supported in 1d. The plane wave (coordinate_system == 1) and planar radial (coordinate_system == -1) versions of the Boussinesq equations are both implemented. The axisymmetric version on the sphere (coordinate_system == 2) is not yet implemented.

Specifying topo and dtopo files is identical to what is described for GeoClaw in One Space Dimension.

**Some things that must change:**

See the examples with names like $CLAW/geoclaw/examples/1d/bouss_* for some sample code that can be modified for other problems.

### Makefile¶

A different Makefile is required for applications to use code from both the $CLAW/geoclaw/src/1d/shallow and $CLAW/geoclaw/src/1d/bouss libraries.

Solving the Boussinesq equations requires solving an elliptic equation each time step, by setting up and solving a tridiagonal linear system of equations. LAPACK is used for this, and the Makefile requires FFLAGS to include -llapack -lblas or explicit pointers to these librarires on your computer. Alternatively, the file $CLAW/geoclaw/src/1d/bouss/lapack_tridiag.f contains the necessary soubroutines from lapack and the blas and so you can add this to the list of SOURCES in the Makefile. See e.g. $CLAW/geoclaw/src/1d/examples/bouss_wavetank_matsuyama/Makefile for an example.

OpenMP is not used in the 1d codes, so there is no need to compile with these flags. For more about FFLAGS and suggested settings for debugging, see FFLAGS environment variable.

### setrun.py¶

Some additional parameters must be added to setrun.py, typically set as follows:

```
from clawpack.geoclaw.data import BoussData1D
rundata.add_data(BoussData1D(),'bouss_data')
rundata.bouss_data.bouss = True
rundata.bouss_data.equations = 2 # for SGN (recommended, or 1 for MS)
rundata.bouss_data.deepBouss = 5. # depth (meters) to switch to SWE
```

The rundata.bouss_data object has attributes:

bouss_equations (int): Which equation set to use (0 for SWE, 1 for MS, 2 for SGN).

bouss_min_depth (float): water depth at which to switch from Boussinesq to SWE.

The latter parameter is needed because in very shallow water, and for modeling onshore inundation, the Boussinesq equations are not suitable. So some criterion is needed to drop these correction terms and revert to solving SWE near shore. Many different approaches have been used in the literature. So far we have only implemented the simplest common approach, which is to revert to SWE in any grid cell where the initial water depth (at the initial time) is less than bouss_min_depth.