Example: spectral_poisson_3d (cfd-3d)
Figure 14.5 — Example: Spectral Poisson 3D
Crate: cfd-3d
Run: cargo run -p cfd-3d --example spectral_poisson_3d
Source: crates/cfd-3d/examples/spectral_poisson_3d.rs
What This Example Demonstrates
The 3D spectral Chebyshev solver on a Poisson problem with a manufactured
analytical solution, verifying exponential convergence of the L2 and L∞ error
as the polynomial-mode count N increases through {4, 6, 8, 10, 12}.
| Aspect | Reference |
|---|---|
| PDE | −∇²u = f on [0,1]³, Dirichlet u = 0 on all faces |
| Source term | f = 3·π²·sin(πx)·sin(πy)·sin(πz) |
| Exact solution | u = sin(πx)·sin(πy)·sin(πz) |
| Discretization | Chebyshev-Gauss-Lobatto nodes, mapped [−1,1] → [0,1] |
| Reference | Canuto et al. (2006), Spectral Methods: Fundamentals in Single Domains, Springer, Ch. 3 |
Key Code Snippet
use cfd_3d::spectral::poisson::PoissonBoundaryCondition;
use cfd_3d::spectral::solver::PoissonProblem;
use cfd_3d::spectral::{SpectralConfig, SpectralSolver};
use leto::Array1;
let source = |x: f64, y: f64, z: f64| -> f64 {
3.0 * std::f64::consts::PI.powi(2) * (std::f64::consts::PI * x).sin()
* (std::f64::consts::PI * y).sin() * (std::f64::consts::PI * z).sin()
};
for &n in &[4usize, 6, 8, 10, 12] {
let config = SpectralConfig::<f64>::new(n, n, n);
let solver = SpectralSolver::new(config);
let problem = PoissonProblem {
source_term: /* covered source */,
bc_x: PoissonBoundaryCondition::Dirichlet(Box::new(|_, _| 0.0)),
bc_y: PoissonBoundaryCondition::Dirichlet(Box::new(|_, _| 0.0)),
bc_z: PoissonBoundaryCondition::Dirichlet(Box::new(|_, _| 0.0)),
};
let sol = solver.solve(&problem)?;
let (l2, linf) = sol.errors_against(&|x, y, z| (PI*x).sin() * (PI*y).sin() * (PI*z).sin());
println!("N = {:>2} | L2 = {:.3e} | Linf = {:.3e}", n, l2, linf);
}Physics Background
A spectral Chebyshev method discretizes a PDE on N+1 Chebyshev-Gauss-Lobatto
points per axis; for smooth (analytic) solutions the L2 error decays
exponentially in N, in contrast to the algebraic O(h^k) decay of
finite-difference schemes. This example constructs a Poisson problem whose
exact answer is a single trigonometric mode, then sweeps N from 4 to 12 and
verifies that the L2 and L∞ errors both decrease monotonically. A runtime
invariant — assert!(l2_err < prev_l2) for every N > 4 — encodes the
monotonic-convergence expectation that the published cfd-3d spectral solver
contract guarantees; a violation surfaces as a panic at the offending N.