Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection–diffusion partial differential equation provides a flexible model class for spatiotemporal processes which is computationally feasible also for large data sets. The Gaussian process defined through the stochastic partial differential equation has, in general, a non-separable covariance structure. Its parameters can be physically interpreted as explicitly modelling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology. To obtain computationally efficient statistical algorithms, we use spectral methods to solve the stochastic partial differential equation. This has the advantage that approximation errors do not accumulate over time, and that in the spectral space the computational cost grows linearly with the dimension, the total computational cost of Bayesian or frequentist inference being dominated by the fast Fourier transform. The model proposed is applied to post-processing of precipitation forecasts from a numerical weather prediction model for northern Switzerland. In contrast with the raw forecasts from the numerical model, the post-processed forecasts are calibrated and quantify prediction uncertainty. Moreover, they outperform the raw forecasts, in the sense that they have a lower mean absolute error.