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Tolosa Documentation

Equations and Discretization

This page presents a simple example of Shallow-Water equations implementation using Tolosa-lib as a tutorial for model development.

Shallow-Water Equations

Section titled “Shallow-Water Equations”

The 2D Shallow-Water equations in non-conservative form:

where:

  • is the water depth
  • is the velocity vector
  • is the gravitational acceleration
  • represents source terms (friction, Coriolis, etc.)

Spatial Discretization

Section titled “Spatial Discretization”

The equations are discretized on unstructured meshes using a finite volume approach.

Cell-centered variables

Section titled “Cell-centered variables”
  • Water depth at cell center
  • Velocities at cell center

Edge-based fluxes

Section titled “Edge-based fluxes”

Fluxes are computed at cell edges using numerical schemes:

where is the maximum wave speed.

Temporal Discretization

Section titled “Temporal Discretization”

Explicit time stepping

Section titled “Explicit time stepping”

Runge-Kutta schemes for time integration:

CFL Condition

Section titled “CFL Condition”

Implementation Structure

Section titled “Implementation Structure”

The Tolosa-lib example implements:

  1. Data structures using type(msh) for mesh
  2. Field arrays for , ,
  3. Flux computation on edges
  4. Time integration with RK schemes
  5. Boundary conditions handling

Source Terms

Section titled “Source Terms”

Bottom friction

Section titled “Bottom friction”

Manning friction:

where is the Manning coefficient.

Coriolis force

Section titled “Coriolis force”

where is the Coriolis parameter.

Numerical Schemes Available

Section titled “Numerical Schemes Available”
  • Upwind schemes: First-order, robust
  • MUSCL schemes: Second-order, less diffusive
  • HLL solver: Riemann solver for fluxes
  • Roe solver: Approximate Riemann solver

Next Steps

Section titled “Next Steps”

See the following pages for implementation details: