Residual-based data-driven variational multiscale reduced order models for parameter-dependent problems

Abstract In this paper, we propose a novel residual-based data-driven closure strategy for reduced order models (ROMs) of under-resolved, convection-dominated problems. The new ROM closure model is constructed in a variational multiscale (VMS) framework by using the available full order model data and a model form ansatz that depends on the ROM residual. We emphasize that this closure modeling strategy is fundamentally different from the current data-driven ROM closures, which generally depend on the ROM coefficients. We investigate the new residual-based data-driven VMS ROM closure strategy in the numerical simulation of three test problems: (i) a one-dimensional parameter-dependent advection-diffusion problem; (ii) a two-dimensional time-dependent advection-diffusion-reaction problem with a small diffusion coefficient ( $$\varepsilon = 1e-4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ε</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> ); and (iii) a two-dimensional flow past a cylinder at Reynolds number $$Re=1000$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mi>e</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1000</mml:mn> </mml:mrow> </mml:math> . Our numerical investigation shows that the new residual-based data-driven VMS-ROM is more accurate than the standard coefficient-based data-driven VMS-ROM.