  Physical models with uncertain inputs are commonly represented as parametric
partial differential equations (PDEs). That is, PDEs with inputs that are
expressed as functions of parameters with an associated probability
distribution. Developing efficient and accurate solution strategies that
account for errors on the space, time and parameter domains simultaneously is
highly challenging. Indeed, it is well known that standard polynomial-based
approximations on the parameter domain can incur errors that grow in time. In
this work, we focus on advection-diffusion problems with parameter-dependent
wind fields. A novel adaptive solution strategy is proposed that allows users
to combine stochastic collocation on the parameter domain with off-the-shelf
adaptive timestepping algorithms with local error control. This is a
non-intrusive strategy that builds a polynomial-based surrogate that is adapted
sequentially in time. The algorithm is driven by a so-called hierarchical
estimator for the parametric error and balances this against an estimate for
the global timestepping error which is derived from a scaling argument.
