  In the task of predicting spatio-temporal fields in environmental science
using statistical methods, introducing statistical models inspired by the
physics of the underlying phenomena that are numerically efficient is of
growing interest. Large space-time datasets call for new numerical methods to
efficiently process them. The Stochastic Partial Differential Equation (SPDE)
approach has proven to be effective for the estimation and the prediction in a
spatial context. We present here the advection-diffusion SPDE with first order
derivative in time which defines a large class of nonseparable spatio-temporal
models. A Gaussian Markov random field approximation of the solution to the
SPDE is built by discretizing the temporal derivative with a finite difference
method (implicit Euler) and by solving the spatial SPDE with a finite element
method (continuous Galerkin) at each time step. The ''Streamline Diffusion''
stabilization technique is introduced when the advection term dominates the
diffusion. Computationally efficient methods are proposed to estimate the
parameters of the SPDE and to predict the spatio-temporal field by kriging, as
well as to perform conditional simulations. The approach is applied to a solar
radiation dataset. Its advantages and limitations are discussed.
