HiPhom𝜺: HIgh order Projection-based HOMogenisation for advection diffusion reaction problems

We propose a new model reduction technique for multiscale scalar transport problems that exhibit dominant axial dynamics. To this aim, we rely on the separation of variables to combine a Hierarchical Model (HiMod) reduction with a two-scale asymptotic expansion. We extend the two-scale asymptotic expansion to an arbitrary order and exploit the high-order correctors to define the HiMod modal basis which approximates the transverse dynamics of the flow, while we adopt a finite element discretisation to model the leading stream. The resulting method, which is named HiPhomε (HIgh-order Projection-based HOMogEnisation), is successfully assessed both in steady and unsteady advection-diffusion-reaction settings. The numerical results confirm the very good performance of HiPhomε which improves the accuracy and the convergence rate of HiMod and extends the reliability of the standard homogenised solution to transient and pre-asymptotic regimes.