# Boussinesq solvers in One Space Dimension¶

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 1d 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 (SWE) 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 second-order derivatives in xx but no time derivatives. In one space dimension, solving this equation requires solving a tridiagonal linear system of equations in each time step.

See Boussinesq solvers in Two Space Dimensions and [BergerLeVeque2023] for more discussion of the Boussinesq-type equations SGN and MS that are implemented in GeoClaw. We recommend using the Serre-Green-Naghdi (SGN) equations rather than Madsen-Sorensen (MS), which has similar dispersive properties but is less stable.

## 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.

The Makefile and setrun.py files must be modified from those used for SWE as described below. See the examples with names like $CLAW/geoclaw/examples/1d/bouss_* for sample files to use as a template.

### 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, which in one space dimension yields 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 libraries 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 $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¶

To use the Boussinesq solvers, somewhere in the setrun function you must include

```
from clawpack.geoclaw.data import BoussData1D
rundata.add_data(BoussData1D(),'bouss_data')
```

and then the following parameters can be adjusted (the values shown here are the default values that will be used if you do not specify a value directly):

```
rundata.bouss_data.equations = 2 # 0=SWE, 1=MS, 2=SGN
rundata.bouss_data.deepBouss = 5. # depth (meters) to switch to SWE
```

The rundata.bouss_data object has attributes:

bouss_equations: The system of equations being solved. Setting this to 2 gives the recommended SGN equations.

The value alpha = 1.153 recommended for SGN is hardwired into $CLAW/geoclaw/src/2d/bouss/bouss_module.f90. To change this value, you must modify this module. (See Library routines in Makefiles for tips on modifying a library routine.) Similarly, if you set bouss_equations = 1 for the Madsen-Sorensen equations, the recommended parameter value Bparam = 1/15 is set in bouss_module.f90.

Setting bouss_equations = 0 causes the code to revert to the shallow water equations, useful for comparing dispersive and nondispersive results.

bouss_min_depth: The criterion used for switching from Boussinesq to SWE in shallow water and onshore. If the original water depth h at time t0 is less than bouss_min_depth in a cell or any of its nearest neighbors, then this cell is omitted from set of unknowns in the elliptic equation solve and no dispersive correction terms are calculated for this cell.

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. See Wave breaking and switching to SWE for more discussion.