Finite Element Solver for the Shallow Water Equations (deal.II)

This repository contains a finite element solver for the Shallow Water Equations (SWE) implemented using the deal.II finite element library.
The software is designed with a dual focus on numerical robustness and parallel scalability.

This is part of a project-work for Numerical Methods For Partial Differential Equations course @Polimi

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Authors

Project developed by:

• Andrea Barletta
• Enrico Tirri

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Overview

The shallow water equations model fluid flows where the horizontal length scales dominate the vertical depth. They are widely used in oceanography, hydraulics, geophysics, and atmospheric sciences.

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Repository Structure

.
├── docs/               # Project documentation
├── gifs/               # Showcase gifs
├── mesh/               # Mesh files
├── scripts/            # Scripts for:
│                           + Plot data
│                           + Generate manufactured solution
│                           + Run tests
├── src/                # C++ source code of the solver
├── CMakeLists.txt      # Build configuration
└── README.md

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Parallelization

The solver is implemented using deal.II's distributed parallel infrastructure:

• parallel::distributed::Triangulation
• MPI communication
• Trilinos linear algebra backends

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Dependencies

• deal.II (compiled with MPI support)
• CMake ≥ 3.16
• MPI (OpenMPI or MPICH)
• C++17 compatible compiler
• Python 3 with numpy and matplotlib (optional)

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Building the Code

From the project root:

mkdir build
cd build
cmake ..
make -j

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Running the Solver

Run in serial:

./shallow_waters <T> <deltat> <mesh_file_name>

Run in parallel:

mpirun -np 4 ./shallow_waters <T> <deltat> <mesh_file_name>

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Manufactured Solutions

The script manufacture.py implements a method of manufactured solutions (MMS) workflow:

• An analytic solution is defined
• Corresponding forcing terms are computed

Run with:

python manufacture.py

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Mesh Generation with Gmsh

The repository includes a geometry definition file mesh/mesh_script.geo that can be used to generate computational meshes using GMSH

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Example Outputs

Below is an example of simulation result.

Gaussian offset bump