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HiPhom$\varepsilon$ -: HIgh order Projection-based HOMogenisation for advection diffusion reaction problems

Giovanni Conni, Stefano Piccardo, Simona Perotto, Giovanni Michele Porta, Matteo Icardi

TL;DR

This work addresses efficient simulation of multiscale scalar transport with dominant axial dynamics by marrying HiMod reduction with high-order two-scale homogenisation. The HiPhom$\varepsilon$ framework constructs problem-specific transverse modes via a higher-order asymptotic expansion and integrates them into a HiMod-like 1D reduction, yielding a rich yet compact reduced model that captures pre-asymptotic and boundary-layer phenomena. Key contributions include a recursive high-order separable expansion, a Gram-Schmidt-orthonormalised transverse modal basis $\{χ_i\}$, and the definition of the enriched space $V_{m,ε}$ that enables offline modal computation and efficient online solution. Numerical tests on Poiseuille and boundary-layer flows show superior accuracy and convergence of HiPhom$\varepsilon$ over standard HiMod, in both steady and unsteady regimes, highlighting its potential for fast, reliable multiscale transport simulations in slender domains. The approach offers a principled, adaptable extension to higher dimensions and more complex boundary conditions, with significant implications for porous media, microfluidics, and networked transport problems.

Abstract

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.

HiPhom$\varepsilon$ -: HIgh order Projection-based HOMogenisation for advection diffusion reaction problems

TL;DR

This work addresses efficient simulation of multiscale scalar transport with dominant axial dynamics by marrying HiMod reduction with high-order two-scale homogenisation. The HiPhom framework constructs problem-specific transverse modes via a higher-order asymptotic expansion and integrates them into a HiMod-like 1D reduction, yielding a rich yet compact reduced model that captures pre-asymptotic and boundary-layer phenomena. Key contributions include a recursive high-order separable expansion, a Gram-Schmidt-orthonormalised transverse modal basis , and the definition of the enriched space that enables offline modal computation and efficient online solution. Numerical tests on Poiseuille and boundary-layer flows show superior accuracy and convergence of HiPhom over standard HiMod, in both steady and unsteady regimes, highlighting its potential for fast, reliable multiscale transport simulations in slender domains. The approach offers a principled, adaptable extension to higher dimensions and more complex boundary conditions, with significant implications for porous media, microfluidics, and networked transport problems.

Abstract

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 (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, 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.
Paper Structure (16 sections, 2 theorems, 58 equations, 19 figures)

This paper contains 16 sections, 2 theorems, 58 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Problem statement and background
  3. HiMod reduction
  4. The domain geometry
  5. The HiMod space
  6. The HiMod reduced problem
  7. Homogenisation via a two-scale asymptotic expansion
  8. High-order asymptotic expansions for axial flows
  9. The HiPhom- method
  10. Numerical assessment
  11. The Poiseuille flow
  12. Convergence analysis
  13. A boundary layer flow
  14. Convergence analysis
  15. The unsteady setting
  16. ...and 1 more sections

Key Result

Proposition 3.1

\newlabellem:theo10 For any $\varepsilon$, the solution $c = c(x,y,t;\varepsilon)$ to the homogenised problem eq:8 does coincide with the asymptotic expansion with $c_0$ the solution to the leading order equation eq:15 and with where function $\chi_i^*$ coincides, up to a constant, with the solution to the boundary value problem being $\chi^*_{-2}=\chi^*_{-1}=0$, $\chi_0^*=1$. In particular, t

Figures (19)

  • Figure 1: Poiseuille flow test case: HiPhom$\varepsilon$ modal basis functions $\chi_i$, for $i=1, \ldots, 4$ (left); comparison among HiPhom$\varepsilon$ and HiMod modal basis functions (right).
  • Figure 2: Poiseuille flow test case: contour plot of the reference solution (top left) and of the HiPhom$\varepsilon$ approximation for $m = 1$ (top-right), and $m=2$, $3$ (bottom, left and right).
  • Figure 3: Poiseuille flow test case: contour plot of the HiMod approximation for $m = 1$, $2$ (top, left and right), and $m=3$, $4$ (bottom, left and right).
  • Figure 4: Poiseuille flow test case: spatial distribution of the absolute modelling error associated with the HiPhom$\varepsilon$ approximation for $m= 1$, $2$ (top, left and right) and $m=3$, $4$ (bottom, left and right).
  • Figure 5: Poiseuille flow test case: spatial distribution of the absolute modelling error associated with the HiMod approximation for $m= 1$, $3$ (top, left and right) and $m=5$, $7$ (bottom, left and right).
  • ...and 14 more figures

Theorems & Definitions (6)

  • Proposition 3.1
  • Proof 1
  • Corollary 3.2: separable representation in an axial flow regime
  • Proof 2
  • Remark 4.1
  • Remark 4.2