Example: richardson_convergence

Figure 16.5 — Example: Richardson Convergence

Crate: cfd-validation Run: cargo run -p cfd-validation --example richardson_convergence Source: crates/cfd-validation/examples/richardson_convergence.rs

What This Example Demonstrates

Richardson extrapolation on the lid-driven cavity benchmark at Re = 100 across three grid levels (16, 32, 64). Estimates the observed order of accuracy p, the extrapolated exact solution, and the Grid Convergence Index (GCI).

Key Code Snippet

use cfd_core::error::Result;
use cfd_validation::benchmarks::{Benchmark, BenchmarkConfig, LidDrivenCavity};

// Re = 100 lid-driven cavity on three uniformly-refined grids.
let grids = [64, 32, 16];           // h1 < h2 < h3 ordering
let f = |n: usize| -> Result<f64> {
    let cavity = LidDrivenCavity::new(1.0, 1.0, 100.0);
    let cfg = BenchmarkConfig::for_grid(n);
    let result = cavity.run(&cfg)?;
    Ok(result.values[0])
};

let (f1, f2, f3) = (f(64)?, f(32)?, f(16)?);
let r = 2.0;
let p = ((f3 - f2) / (f2 - f1)).abs().ln() / r.ln();
let f_rich = f1 + (f1 - f2) / (r.powf(p) - 1.0);
let gci = 1.25 * (f1 - f2).abs() / f1 / (r.powf(p) - 1.0);

Physics Background

Richardson extrapolation (Richardson & Glauert, 1927; later formalized by Roache as the Grid Convergence Index) combines solutions from three uniformly-refined grids to estimate both the observed order of accuracy and the continuum value the finite-difference solution is converging toward. The GCI with the safety factor 1.25 (Roache, 1994) provides a conservative asymptotic bound on the discretization error; for a formally second-order scheme on a smooth benchmark such as lid-driven cavity at Re = 100, the observed p should land near 2, and a small GCI certifies that the resolved grids lie in the asymptotic regime. This example serves as CFDrs’s code-verification oracle: a dip in p flags implementation drift.

Generated Figure

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