A fast Chebyshev exponential-integrator solver for stochastic PDEs on bounded domains with non-homogeneous Dirichlet...

We present a Chebyshev-collocation exponential-integrator scheme for parabolic stochastic partial differential equations (SPDEs) on a bounded interval with time-dependent non-homogeneous Dirichlet boundary data, where periodic-domain Fourier methods and standard deterministic solvers do not apply. The method combines Chebyshev-Gauss-Lobatto differentiation matrices for the spatial Laplacian; a stochastic exponential-Euler time step whose linear dynamics are propagated exactly via matrix φ-functions computed by Talbot-contour quadrature and whose stochastic convolution is sampled exactly in the operator eigenbasis; and a Clenshaw-Curtis noise-scaling rule that gives the correct continuum white-noise limit on the non-uniform grid. We validate the scheme for additive space-time white noise and temporally coloured Ornstein-Uhlenbeck noise, and benchmark it against semi-implicit Euler-Maruyama, a stabilized explicit Runge-Kutta-Chebyshev method (S-ROCK), and explicit Euler-Maruyama. On a linear benchmark with known stationary variance, the method reaches the Monte-Carlo floor (0.93% peak relative error over 120 realisations) and stays there across two decades of time step, while the competitors degrade. Temporal convergence is order one and spatial convergence is spectral. On the stochastic Allen-Cahn equation the scheme is 7.5 × faster than a fine-step semi-implicit Euler-Maruyama reference at matched accuracy, and on the stiff stochastic Burgers equation it remains stable across an 8 × wider time-step range than semi-implicit Euler-Maruyama.