Crate numra_spde 

Stochastic Partial Differential Equation (SPDE) solvers for Numra.

This crate provides solvers for SPDEs of the form:

∂u/∂t = L[u] + σ(u) ξ(x,t)

where L is a spatial differential operator, σ(u) is the diffusion coefficient, and ξ(x,t) is space-time white noise.

Method

We use the Method of Lines (MOL) approach:

1. Discretize the spatial operator L using finite differences
2. Convert to a system of SDEs
3. Solve using SDE solvers from numra-sde

Example: Stochastic Heat Equation

use numra_spde::{SpdeSystem, SpdeSolver, SpdeOptions, SpdeResult};
use numra_pde::Grid1D;
use numra_core::Scalar;

// Stochastic heat equation: ∂T/∂t = α ∂²T/∂x² + σ dW(x,t)
struct StochasticHeat {
    alpha: f64,  // Thermal diffusivity
    sigma: f64,  // Noise intensity
}

impl SpdeSystem<f64> for StochasticHeat {
    fn dim(&self) -> usize { 1 }  // 1D spatial

    fn drift(&self, _t: f64, u: &[f64], du: &mut [f64], grid: &Grid1D<f64>) {
        let dx = grid.dx_uniform();
        let n = u.len();
        for i in 0..n {
            let u_left = if i == 0 { 0.0 } else { u[i - 1] };  // Dirichlet BC
            let u_right = if i == n - 1 { 0.0 } else { u[i + 1] };  // Dirichlet BC
            du[i] = self.alpha * (u_left - 2.0 * u[i] + u_right) / (dx * dx);
        }
    }

    fn diffusion(&self, _t: f64, u: &[f64], sigma: &mut [f64], _grid: &Grid1D<f64>) {
        for i in 0..u.len() {
            sigma[i] = self.sigma;  // Additive noise
        }
    }
}

Author: Moussa Leblouba Date: 3 February 2026 Modified: 2 May 2026

Structs

ColoredNoise

Spatially colored (correlated) noise.

ColoredNoiseGenerator

Spatially correlated Gaussian noise generator.

Grid1D

1D uniform or non-uniform grid.

MolSdeSolver

Method of Lines SPDE solver.

SpdeEnsemble

Ensemble solver for SPDE Monte Carlo simulations.

SpdeOptions

Options for SPDE solver.

SpdeResult

Result of SPDE solve.

SpdeStats

Statistics for SPDE solve.

WhiteNoise

Spatially white (uncorrelated) space-time noise.

Enums

NoiseCorrelation

Noise correlation structure.

SpdeMethod

SDE integration method for the spatially-discretized system.

Traits

Scalar

A real scalar type suitable for numerical computation.

SpaceTimeNoise

Trait for space-time noise processes.

SpdeSolver

Trait for SPDE solvers.

SpdeSystem

Trait for Stochastic Partial Differential Equation systems.