Data‐Driven Optimization for the Evolve‐Filter‐Relax Regularization of Convection‐Dominated Flows

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 that consists of 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 (Formula presented.) and the relaxation parameter (Formula presented.). Standard choices for these parameters are usually fixed in time, and related to the full order model setting, that is, the grid size for (Formula presented.) and the time step for (Formula presented.). The key novelties with respect to the standard EFR approach are: (i) time-dependent parameters (Formula presented.) and (Formula presented.), 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 (Formula presented.). The Opt-EFR proved to be more accurate than standard approaches by up to 99 (Formula presented.), 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.