Struct FDM 

pub struct FDM;

Finite Difference Method utilities.

Implementations

impl FDM

pub fn d1_central<S: Scalar>(u: &[S], dx: S, i: usize) -> S

Compute first derivative using central differences.

du/dx ≈ (u[i+1] - u[i-1]) / (2*dx)

pub fn d1_forward<S: Scalar>(u: &[S], dx: S, i: usize) -> S

Compute first derivative using forward difference.

du/dx ≈ (u[i+1] - u[i]) / dx

pub fn d1_backward<S: Scalar>(u: &[S], dx: S, i: usize) -> S

Compute first derivative using backward difference.

du/dx ≈ (u[i] - u[i-1]) / dx

pub fn d2_central<S: Scalar>(u: &[S], dx: S, i: usize) -> S

Compute second derivative using central differences.

d²u/dx² ≈ (u[i+1] - 2*u[i] + u[i-1]) / dx²

pub fn d2_central4<S: Scalar>(u: &[S], dx: S, i: usize) -> S

Compute second derivative using fourth-order central differences.

d²u/dx² ≈ (-u[i+2] + 16*u[i+1] - 30*u[i] + 16*u[i-1] - u[i-2]) / (12*dx²)

pub fn d1_left_boundary<S: Scalar>(u: &[S], dx: S) -> S

Compute first derivative at left boundary using one-sided differences.

du/dx ≈ (-3*u[0] + 4*u[1] - u[2]) / (2*dx)

pub fn d1_right_boundary<S: Scalar>(u: &[S], dx: S) -> S

Compute first derivative at right boundary using one-sided differences.

du/dx ≈ (3*u[n-1] - 4*u[n-2] + u[n-3]) / (2*dx)

pub fn laplacian_1d<S: Scalar>(u: &[S], dx: S) -> Vec<S>

Apply Laplacian operator to entire field (interior points only).

Returns vector of size n-2 (excluding boundary points).

pub fn laplacian_2d<S: Scalar>( u: &[S], nx: usize, ny: usize, dx: S, dy: S, ) -> Vec<S>

Apply Laplacian operator to entire 2D field (interior points only).

Uses a 5-point stencil.

pub fn gradient_1d<S: Scalar>(u: &[S], dx: S) -> Vec<S>

Compute gradient at all interior points.

pub fn laplacian_matrix_1d<S: Scalar>(n_interior: usize, dx: S) -> Vec<Vec<S>>

Build sparse Laplacian matrix for 1D domain with given BCs.

Returns a tridiagonal matrix in dense form for simplicity. For large problems, use sparse matrix formats.