Data-driven Optimization for the Evolve-Filter-Relax regularization of convection-dominated flows

TL;DR

This paper introduces a data-driven, adaptive optimization approach for the evolve-filter-relax (EFR) method to improve the accuracy of convection-dominated flow simulations, achieving up to 99% better accuracy without extra computational cost.

Contribution

It proposes time-dependent, data-driven optimization of EFR parameters, incorporating spatial gradient information and global metrics, which enhances simulation accuracy over standard fixed-parameter methods.

Findings

Optimized EFR strategies significantly improve accuracy in turbulent flow simulations.

Including spatial gradients and global metrics in the objective function is crucial for optimal performance.

The approach maintains computational efficiency comparable to standard methods.

Abstract

Numerical stabilization techniques are often employed in under-resolved simulations of convection-dominated flows to improve accuracy and mitigate spurious oscillations. Specifically, the evolve--filter--relax (EFR) algorithm is a framework which consists in evolving the solution, applying a filtering step to remove high-frequency noise, and relaxing through a convex combination of filtered and original solutions. The stability and accuracy of the EFR solution strongly depend on two parameters, the filter radius $\delta$ and the relaxation parameter $\chi$. Standard choices for these parameters are usually fixed in time, and related to the full order model setting, i.e., the grid size for $\delta$ and the time step for $\chi$. The key novelties with respect to the standard EFR approach are: (i) time-dependent parameters $\delta(t)$ and $\chi(t)$, and (ii) data-driven adaptive optimization of the parameters in time, considering a fully-resolved simulation as reference. In particular, we propose three different classes of optimized-EFR (Opt-EFR) strategies, aiming to optimize one or both parameters. The new Opt-EFR strategies are tested in the under-resolved simulation of a turbulent flow past a cylinder at $Re=1000$. The Opt-EFR proved to be more accurate than standard approaches by up to 99$\%$, while maintaining a similar computational time. In particular, the key new finding of our analysis is that such accuracy can be obtained only if the optimized objective function includes: (i) a global metric (as the kinetic energy), and (ii) spatial gradients' information.