Matrix-Free Operators and Krylov Solvers

Figure 11.1 — Matrix-Free Operators and Krylov Solvers

For high-resolution flows, CFDrs prefers matrix-free evaluation: the linear operator is a trait, and the solver allocates only the action function A·x. Matrix-free applications decouple memory footprint from discretization size.

pub trait LinearOp<F: FloatElement> {
    fn apply(&self, x: &NdArray<F, Ix3>, y: &mut NdArray<F, Ix3>);
}

A Krylov solver asks only for apply, so it composes with any of the three discretizations (Stencil, FEM, Spectral).

Iterative Solvers

cfd-math::krylov exposes BiCGSTAB, GMRES, and IDR(s). All three reach the same convergence target at the same iteration count regardless of which discretization backs them.

let krylov = BiCgStab::<f64>::new();
let solution = krylov.solve(&op, &rhs, &preconditioner, 1e-6, 5000)?;

Examples Referenced by This Chapter

• matrix_free_demo — matrix-free operator application paired with a Krylov solver.

Further Reading

• For spatial discretization surfaces, time integration, and adaptive time-stepping, see Spectral, FEM, MUSCL, and Matrix-Free Methods.
• cfd-math source