HiPhom$\varepsilon$ -: 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 HiPhom$\varepsilon$ (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 HiPhom$\varepsilon$, 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.