# Boussinesq solvers in Two Space Dimensions¶

As of Version 5.10.0, GeoClaw includes the option to solve a dispersive Boussinesq-type equation known as Serre-Green-Naghdi (SGN) instead of the usual shallow water equations (SWE). This equation and related depth-averaged equations have been used extensively in the literature to better model wave propagation in situations where the wavelength is not sufficiently long relative to the fluid depth for the SWE approximation to be accurate. Applications include the study of tsunamis generated by landslides, asteroid impacts, or other localized phenomena. Including dispersive terms has also been found to give more realistic results for earthquake-generated tsunamis in some situations. See [BergerLeVeque2023] for references to some of this literature along with more discussion of the GeoClaw implementation and test problems.

The one-dimensional version of these capabilities are described in Boussinesq solvers in One Space Dimension.

The SGN equations are still depth-averaged equations, with the same conserved quantities (fluid depth h and momenta hu and hv in 2D) as the shallow water equations (SWE), but the equations contain higher order derivative terms and so they are no longer hyperbolic and require implicit methods for efficient solution with a physically reasonable time step. This adds considerable complexity to the code since adaptive mesh refinement (AMR) is still supported. The implementation proceeds by alternating time steps on the shallow water equations (SWE) with the solution of elliptic equations where the operator involves second order derivatives in x and y of a new set of variables used to modify the momenta each time step. The right hand side also involves third order derivatives of the topography.

In two space dimensions, solving this elliptic equation requires setting up and solving a sparse linear system of equations in each time step, at each refinement level when AMR is being used. All grid cells from all patches at the same refinement level are included in the linear system. Boundary conditions at the edge of patches must be interpolated from coarser level solutions, in much the same way that the boundary conditions for h, hu, and hv are interpolated when solving the SWE with AMR. Because the solution of the elliptic system yields correction terms to the momenta (denoted here by huc and hvc), when solving the Boussinesq equations the array q of state variables is expanded to include these correction terms as well, and so the number of equations and variables is 5 instead of 3. This change must be made in setrun.py, along with other changes discussed below, in order to use the Boussinesq solvers.

Currently the only linear system solver supported for solving the large sparse elliptic systems is PETSc, which can use MPI to solve these these systems. Using the Boussinesq solvers requires these prerequesites, as discussed further in Prerequisites for the 2d Boussinesq code.

See [BergerLeVeque2023] for more discussion of the equations solved and the AMR algorithms developed for these equations.

## Boussinesq-type dispersive equations¶

The equations we solve are not the original depth-averaged dispersive equations derived by Boussinesq, but for simplicity in this documentation and the code, we often refer to the equations simply as “Boussinesq equations”, following common practice. Many variants of these equations have been derived and fine-tuned to better match the dispersion relation of the linearized Airy wave theory and to incorporate variable bottom topography.

Two variants are currently implement in GeoClaw, described below. In practice we recommend using only the SGN equations, which we have found to be more stable.

## The SGN equations¶

The Serre-Green-Naghdi (SGN) equations implemented in GeoClaw are generalized to include 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]. The value 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. This value 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.) Setting alpha = 1 gives the original SGN equations.

## The Madsen-Sorensen (MS) equations¶

Primarily for historical reasons, GeoClaw also includes an implementation of another Boussinesq-type dispersive system, the Madsen-Sorensen (MS) equations. These equations also give a good approximation to the linear dispersion relation of the Airy solution when the parameter Bparam = 1/15 is used, which is hardwired into $CLAW/geoclaw/src/2d/bouss/bouss_module.f90. These equations were used in an earlier GeoClaw implementation by Jihwan Kim, known as BoussClaw [KimEtAl2017]. This implementation was successfully used in a number of studies (see [BergerLeVeque2023] for some citations). However, extensive tests with these equations have revealed stability issues, particularly with the use of AMR, which have also been reported by others. Implementations of MS in both 1D and 2D are included in GeoClaw, but are generally not recommended except for those interested in comparing different formulations for small numbers of time steps, and perhaps further investigating the stability issues.

## Using the 2d Boussinesq code¶

Provided the Prerequisites for the 2d Boussinesq code have been installed, switching from the SWE to a Boussinesq solver in GeoClaw requires only minor changes to setrun.py and the Makefile.

See the files in $CLAW/geoclaw/examples/bouss/radial_flat for an example.

### setrun.py¶

As mentioned above, it is necessary to set:

```
clawdata.num_eqn = 5
```

instead of the usual value 3 used for SWE, since correction terms for the momenta are also stored in the q arrays to facilitate interpolation to ghost cells of finer level patches each time step.

It is also necessary to set some parameters that are specific to the Boussinesq solvers. Somewhere in the setrun function you must include

```
from clawpack.geoclaw.data import BoussData
rundata.add_data(BoussData(),'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.bouss_equations = 2 # 0=SWE, 1=MS, 2=SGN
rundata.bouss_data.bouss_min_level = 1 # coarsest level to apply bouss
rundata.bouss_data.bouss_max_level = 10 # finest level to apply bouss
rundata.bouss_data.bouss_min_depth = 10. # depth (meters) to switch to SWE
rundata.bouss_data.bouss_solver = 3 # 3=PETSc
rundata.bouss_data.bouss_tstart = 0. # time to switch from SWE
```

These parameters are described below:

bouss_equations: The system of equations being solved. Setting this to 2 gives the recommended SGN equations, while 1 gives Madsen-Sorensen.

Setting bouss_equations = 0 causes the code to revert to the shallow water equations, useful for comparing dispersive and nondispersive results. (But if bouss_data is being set, it still requires clawdata.num_eqn = 5 and the two new components in q are always 0 in this case, so this is slightly less efficient than using the standard GeoClaw.)

bouss_min_level: The minimum AMR level on which Boussinesq correction terms should be applied. In some cases it may be desirable to use the SWE on the coarsest grids in the ocean while Boussinesq corrections are only applied on fine levels near shore, for example.

bouss_max_level: The finest AMR level on which Boussinesq correction terms should be applied. In some cases it may be desirable to use the SWE only on coarser grids if the finest level grid only exists in very shallow regions or onshore, where the the equations switch to SWE for inundation modeling. Since much of the computational work is often on the finest level, avoiding the Boussinesq terms altogether on these levels may be advantageous in some situations.

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 in a 3-by-3 neighborhood, then this cell is omitted from set of unknowns in the elliptic equation solve and no dispersive correction terms are calculated for this cell. This is discussed further below in Wave breaking and switching to SWE.

bouss_solver: What linear system solver to use. Currently only the value 3 for PETSc is recognized.

bouss_tstart: The time t at which to start applying Boussinesq terms. Normally you will want this to be less than or equal to t0, the starting time of the calculation (which is not always 0). However, there are some cases in which the initial data results in extreme motion in the first few time steps and it is necessary to get things going with the SWE. For most applications this is not necessary and you need only change this parameter if you are solving a problem for which t0 < 0.

### Makefile¶

You can copy the Makefile from $CLAW/geoclaw/examples/bouss/radial_flat/Makefile and make any adjustments needed.

This Makefile reads in the standard Boussinesq solver file $CLAW/geoclaw/src/2d/bouss/Makefile.bouss, which lists the Fortran modules and source code files that are used by default from the library $CLAW/geoclaw/src/2d/bouss, or from $CLAW/amrclaw/src/2d or $CLAW/geoclaw/src/2d/shallow in the case of files that did not need to be modified for the Boussinesq code.

Two Makefile variables PETSC_DIR and PETSC_ARCH must be set (perhaps as environment variables in the shell from which make is invoked). These are described further below in Prerequisites for the 2d Boussinesq code.

The FFLAGS specified in the Makefile should include -DHAVE_PETSC to indicate that PETSc is being used, necessary when compiling the source code for Boussinesq solvers.

The Makefile should also include a line of the form:

```
PETSC_OPTIONS=-options_file $(CLAW)/geoclaw/examples/bouss/petscMPIoptions
```

with a pointer to the file that sets various PETSc options. The file $CLAW/geoclaw/examples/bouss/petscMPIoptions gives the options used in the examples, which may be adequate for other problems too. This file includes some comments briefly explaining the options set. We use a GMRES Krylov space method as the main solver and algebraic multigrid as the preconditioner. For more about the options for these methods, see:

In addition to a line of the form

```
EXE = xgeoclaw
```

that specifies the name and location of the executable to be generated, the Makefile should also contain a line of the form:

```
RUNEXE="${PETSC_DIR}/${PETSC_ARCH}/bin/mpiexec -n 6"
```

This is the command that should be used in order to run the executable. In other words, if you set PETSC_DIR and PETSC_ARCH as environment variables, and the executable is named xgeoclaw as usual, then the command

```
$PETSC_DIR/$PETSC_ARCH/bin/mpiexec -n 6 xgeoclaw
```

given in the shell should run the executable (invoking MPI with 6 processes in this example). If this does not work then one of the environment variables may be set incorrectly to find the mpiexec command.

## Prerequisites for the 2d Boussinesq code¶

Currently the only linear solver supported is PETSc, so this must be installed, see https://petsc.org/release/install/ for instructions and also note the PETSc prerequisites. Note that MPI, LAPACK, and the BLAS are required and will be installed as part of installing PETSc. If you already have some of the prerequisites installed, be sure to read Configuring PETSc before installing.

The environment variables $PETSC_DIR and $PETSC_ARCH must be set appropriately based on your PETSc installation, either as environment variables or directly in the Makefile. See the PETSc documentation page Environmental Variables $PETSC_DIR And $PETSC_ARCH.

## Wave breaking and switching to SWE¶

The bouss_min_depth 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 commonly used 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.

## Examples¶

In addition to one example application included in GeoClaw, found in the directory $CLAW/geoclaw/examples/bouss/radial_flat, several other examples of usage can be found in the code repository https://github.com/rjleveque/ImplicitAMR-paper, which was developed to accompany the paper [BergerLeVeque2023].