Kinetic-Solver

A finite volume solver for the one-dimensional Saint-Venant (shallow water) system with topography, built around the kinetic scheme with reflections of B. Perthame and C. Simeoni (Calcolo, 2001).

\[ \partial_t h + \partial_x (hu) = 0, \qquad \partial_t (hu) + \partial_x\!\left(hu^2 + \tfrac{g}{2}h^2\right) = -g\,h\,\partial_x Z . \]

The scheme treats the bottom \(Z\) as a potential barrier for fictitious particles: particles cross a step when they have enough kinetic energy and are reflected otherwise. This gives a finite volume method that

• keeps the water height non-negative, dry areas included;
• preserves the lake at rest (\(u = 0\), \(h + Z\) constant) exactly;
• satisfies a discrete entropy inequality;
• stays stable on discontinuous bottoms.

A Rusanov scheme with hydrostatic reconstruction (Audusse et al., 2004) is included for comparison.

In thirty seconds

make
./build/kinetic-solver --case dam-break-wet --cells 400 --snapshot-every 20
python3 scripts/plot.py output --animate
make test

Where to go next

Page	What you will find
Getting started	Building, running, options, output files, plots
Theory	The equations, the kinetic formulation and the scheme of the paper
Implementation	How the code is organised and the numerical choices it makes
Validation	Exact solutions, convergence, well-balancing, symmetries
Contributing	Tests, adding a scheme, a case or an integrator
References	Papers the code is based on

Status

The scheme of the paper is implemented and checked against its definitions to round-off (see Validation). The code was first written in 2013; the 2026 revision fixed several bugs (including a truncated value of π that broke the lake-at-rest property), restructured the code and added a test suite. The original guide, Guidebook.pdf in the repository, describes the 2013 version.