Arbitrary Lagrangian Eulerian (ALE) formulations are useful when dealing with problems defined on moving domains, such as fluid structure interactions. Having at hand higher order (at least second order) in time ALE methods is important, as they lead to realistic simulations involving fluids in 3d. However such methods are very limited in the literature.

In this talk we propose and analyse higher order in time ALE methods for an advection–diffusion model problem defined on time–dependent domains. In particular, we discretize only in time by the discontinuous Galerkin (dG) method and, by involving appropriate quadratures, we propose ALE methods of any order in time. The proposed methods enjoy the same stability properties as the continuous problem. Our approach is a generalisation of the so-called Geometric Conservation Law (GCL). We also provide a priori and a posteriori error analyses. The a priori error analysis is based on the introduction of a novel time–space projection, the so-called “ALE projection”, while the a posteriori error analysis is based on the reconstruction technique and the definition of a novel time–space reconstruction.