Advection

apply_scalar_advection and apply_vector_advection implement first-order upwind advection for scalar and vector (1-form) fields on a discrete mesh.

Summary

Upwind Scheme

The upwind direction for edge is determined by the sign of the velocity projection onto the edge orientation:

This is the discrete analogue of : the directional derivative of along the velocity field .

Code Examples

Scalar advection

use cartan_dec::{Mesh, apply_scalar_advection};
use nalgebra::DVector;
 
let mesh = Mesh::unit_square_grid(4);
 
// Scalar field on vertices
let f = DVector::from_fn(mesh.n_vertices(), |i, _| {
    let [x, _y] = mesh.vertices[i];
    x   // linear field f = x
});
 
// Velocity field on edges (1-form)
let u = DVector::from_element(mesh.n_edges(), 0.5_f64);  // uniform flow
 
let df_dt = apply_scalar_advection(&mesh, &f, &u);
assert_eq!(df_dt.len(), mesh.n_vertices());

Time stepping

let dt = 0.01_f64;
let f_next = &f + dt * apply_scalar_advection(&mesh, &f, &u);

Vector advection

use cartan_dec::apply_vector_advection;
 
// 1-form alpha on edges
let alpha = DVector::from_element(mesh.n_edges(), 1.0_f64);
let dalpha_dt = apply_vector_advection(&mesh, &alpha, &u);
assert_eq!(dalpha_dt.len(), mesh.n_edges());

Numerical Notes

CFL condition: the upwind scheme is stable when . For a uniform grid with edge length and maximum velocity , use .

Upwind direction: determined by the sign of the velocity projection on each edge. Zero velocity on an edge contributes no advective flux at that edge.

Applications

• Active nematics: advection of Q-tensor field by active flow in volterra
• Scalar transport: tracer advection in fluid simulations
• Material lines: passive scalar advection for flow visualization