# Shallow water model for year-round watershed runoff

I have an (maybe not so clever) idea to apply the shallow water model for computing the year-round watershed runoff of a catchment. It means using of real topography with variable slopes and roughness, length of the real time period about 31536000 seconds, real precipitation, temperature, snow cover and other data, etc. I read some books and papers about hydrodynamic models and many, many numerical methods for solving them. Shallow water model is usually applied for oceanic circulation, tides and estuaries, flood waves with relatively short time duration, etc. There are plenty of methods, described in the literature, like explicit and implicit ones, finite difference, finite volume, finite element, and so on. Every class of methods has its own pros and cons. There are also different flow regimes, different types of initial and boundary conditions, different conditions of the flow over the bottom, for example wet/dry conditions, etc. Not to mention some other processes and models like infiltration and porous media model which are also natural (but at the beginning only the shallow water model is important).

So, the questions are:

1. Is this idea even applicable and doable (this is especially about year-round duration of the time period)?
2. Which numerical methods are simple and stable, for starting the project, solving the shallow water model, and obtaining reasonable results at the beginning (which means positive water height, oscillations as small as possible, etc.)?
3. Which numerical methods are not so simple but yet stable, for advanced modelling, and obtaining better results with the shallow water model?

I appreciate every comment, suggestion, and advise.

• Why not use something that already does this, like Delft3D oss.deltares.nl/web/delft3d ? Jul 6, 2023 at 14:07
• Most 1D channel flow models are shallow water based, usually with terms added for cross-section geometry. In other words, they integrate across the channel. There are a number of other approximations then applied depending on the situation. The reason why SWE are not used across large areas is that the amount of water flowing across and into channels is small and tends to be difficult to capture numerically. Delft3D can be used but is very expensive and only used primarily in coastal areas. Common models: hec.usace.army.mil/confluence/hmsdocs/hmstrm/channel-flow Jul 7, 2023 at 13:04
• There is a whole NSF-funded center devoted to modeling of surface processes, including dealing with rain water and ground water and how they erode the surface: csdms.colorado.edu/wiki/Main_Page Jul 7, 2023 at 23:31
• The CSDMS annual meeting is a lot of fun! And very nicely sized. Sep 23, 2023 at 15:28

## 2 Answers

I don't have a lot of experience in fluid simulation. But here are my answers to your questions:

# Question 1: Is this idea even applicable and doable (this is especially about year-round duration of the time period)?

The short answer is yes, but there are a lot of things to take into account.

## Theory/Physical model

You mentioned multiple physical phenomena. Each one of them has a system of equations and multiple terms within each equation. You can improve the equations by adding new terms. Also you have to take care of the coupling between the phenomena. Even if your model gives poor results, maybe adding another set of equations will improve the results and vice-versa.

## Numerical methods to solve the equations

Alongside your intuition, you can study the sets of equations and choose a proper scheme. As you have read, the finite volume method (FV) is the most used for conservation laws.

## Computational resources

This part also depends on how much data you have (topography). Once your model run in an acceptable amount of time. You can add small tweaks to have better results and test other schemes. For example, if a single time-step takes too much time in explicit. Maybe using an implicit scheme is the way to go to reduce the number of time-steps. It doesnt change the fact that you should profile your code, so that your entire simulation has an acceptable running time.

# Question 2: Which numerical methods are simple and stable, for starting the project, solving the shallow water model, and obtaining reasonable results at the beginning (which means positive water height, oscillations as small as possible, etc.)?

The first thing that comes to my mind is the finite volume method. The choice of the numerical flux are the Rusanov or Harten-Lax-van Leer. These fluxes are diffusive and are positive (keep the height positive).

# Question 3: Which numerical methods are not so simple but yet stable, for advanced modelling, and obtaining better results with the shallow water model?

There are multiple scheme you can try (this list is not exhaustive):

• High order fluxes
• Limiter schemes
• predictor-corrector schemes
• MUSCL schemes
• etc
• Linking to relevant literature does not constitute a complete answer. Oct 23, 2023 at 15:49