An Adaptive Time Filtered Backward Euler Method for Reduced-Order Models of Incompressible Flows

This paper studies a reduced-order model based on proper orthogonal decomposition (POD) for the incompressible Navier-Stokes equations. The difficulties resulting from nonlinearity are eliminated using the variational multiscale (VMS) method. The time filter is added as a separate post-processing step to the standard VMS-POD approximation. This increases the accuracy and presents a better energy preserving scheme without adding additional computational complexity. The stability and error analyses of the method are provided, and results of the several numerical tests are presented to verify the efficiency of the method in this setting.

Keywords:

Proper orthogonal decomposition, Time filter, Reduced-order models Projection-based variational multiscale,

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[1] Eroglu, F., Kaya, S., Rebholz, L., “POD-ROM for the Darcy-Brinkman equations with double-diffusive convection”, Journal of Numerical Mathematics, 27(3): 123–139, (2019).
[1] Eroglu, F., Kaya, S., Rebholz, L., “POD-ROM for the Darcy-Brinkman equations with double-diffusive convection”, Journal of Numerical Mathematics, 27(3): 123–139, (2019).
[2] Kunisch, K., Volkwein, S., “Galerkin proper orthogonal decomposition methods for parabolic problems”, Numerische Mathematik, 90: 117–148, (2001).
[2] Kunisch, K., Volkwein, S., “Galerkin proper orthogonal decomposition methods for parabolic problems”, Numerische Mathematik, 90: 117–148, (2001).
[3] San, O., Borggaard, J., “Principal interval decomposition framework for POD reduced-order modeling of convective Boussinesq flows”, International Journal for Numerical Methods in Fluids, 78(1): 37–62, (2015).
[3] San, O., Borggaard, J., “Principal interval decomposition framework for POD reduced-order modeling of convective Boussinesq flows”, International Journal for Numerical Methods in Fluids, 78(1): 37–62, (2015).
[4] Eroglu, F., Kaya, S., Rebholz, L., “A modular regularized variational multiscale proper orthogonal decomposition for incompressible flows”, Computer Methods in Applied Mechanics and Engineering, 325: 350–368, (2017).
[4] Eroglu, F., Kaya, S., Rebholz, L., “A modular regularized variational multiscale proper orthogonal decomposition for incompressible flows”, Computer Methods in Applied Mechanics and Engineering, 325: 350–368, (2017).
[5] Eroglu, F., Kaya, S., Rebholz, L., “Decoupled modular regularized VMS-POD for Darcy-Brinkman equations”, IAENG International Journal of Applied Mathematics, 49(2): 134–144, (2019).
[5] Eroglu, F., Kaya, S., Rebholz, L., “Decoupled modular regularized VMS-POD for Darcy-Brinkman equations”, IAENG International Journal of Applied Mathematics, 49(2): 134–144, (2019).
[6] Iliescu, T., Wang Z., “Variational multiscale proper orthogonal decomposition: Navier-Stokes equations”, Numerical Methods for Partial Differential Equations, 30(2): 641–663, (2014).
[6] Iliescu, T., Wang Z., “Variational multiscale proper orthogonal decomposition: Navier-Stokes equations”, Numerical Methods for Partial Differential Equations, 30(2): 641–663, (2014).
[7] Iliescu, T., Wang, Z., “Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations”, Mathematics of Computation, 82(283): 1357–1378, (2013).
[7] Iliescu, T., Wang, Z., “Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations”, Mathematics of Computation, 82(283): 1357–1378, (2013).
[8] Roop, J. P., “A proper-orthogonal decomposition variational multiscale approximation method for a generalized Oseen problem”, Advances in Numerical Analysis, 2013: 1–8, (2013).
[8] Roop, J. P., “A proper-orthogonal decomposition variational multiscale approximation method for a generalized Oseen problem”, Advances in Numerical Analysis, 2013: 1–8, (2013).
[9] Cibik, A., Demir, M., Kaya, S., “A family of second order time stepping methods for the Darcy-Brinkman equations”, Journal of Mathematical Analysis and Applications, 472(1): 148–175, (2019).
[9] Cibik, A., Demir, M., Kaya, S., “A family of second order time stepping methods for the Darcy-Brinkman equations”, Journal of Mathematical Analysis and Applications, 472(1): 148–175, (2019).
[10] Guzel, A., Layton, W., “Time filters increase accuracy of the fully implicit method”, BIT Numerical Mathematics, 58: 301–315, (2018).
[10] Guzel, A., Layton, W., “Time filters increase accuracy of the fully implicit method”, BIT Numerical Mathematics, 58: 301–315, (2018).
[11] Akbas, M., “An adaptive time filter based finite element method for the velocity-vorticity-temperature model of the incompressible non-isothermal fluid flows”, Gazi University Journal of Science, 33(3): 696–713, (2020).
[11] Akbas, M., “An adaptive time filter based finite element method for the velocity-vorticity-temperature model of the incompressible non-isothermal fluid flows”, Gazi University Journal of Science, 33(3): 696–713, (2020).
[12] Cibik, A., Eroglu, F., Kaya, S., “Analysis of second order time filtered backward Euler method for MHD equations”, Journal of Scientific Computing, 82: 38, (2020).
[12] Cibik, A., Eroglu, F., Kaya, S., “Analysis of second order time filtered backward Euler method for MHD equations”, Journal of Scientific Computing, 82: 38, (2020).
[13] DeCaria, V., Layton, W., Zhao, H., “A time-accurate, adaptive discretization for fluid flow problems”, International Journal of Numerical Analysis and Modeling, 17(2): 254–280, (2020).
[13] DeCaria, V., Layton, W., Zhao, H., “A time-accurate, adaptive discretization for fluid flow problems”, International Journal of Numerical Analysis and Modeling, 17(2): 254–280, (2020).
[14] Adams, R., “Sobolev spaces”, Academic Press, Newyork, (1975).
[14] Adams, R., “Sobolev spaces”, Academic Press, Newyork, (1975).
[15] Layton, W., “Introduction to the numerical analysis of incompressible viscous flows”, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA, (2008).
[15] Layton, W., “Introduction to the numerical analysis of incompressible viscous flows”, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA, (2008).
[16] Jiang, N., Mohebujjaman, M., Rebholz, L., Trenchea, C., “An optimally accurate discrete regularization for second order timestepping methods for Navier-Stokes equations”, Computer Methods in Applied Mechanics and Engineering, 310: 388–405, (2016).
[16] Jiang, N., Mohebujjaman, M., Rebholz, L., Trenchea, C., “An optimally accurate discrete regularization for second order timestepping methods for Navier-Stokes equations”, Computer Methods in Applied Mechanics and Engineering, 310: 388–405, (2016).
[17] Trenchea, C., “Second-order unconditionally stable IMEX schemes: Implicit for local effects and explicit for nonlocal effects”, ROMAI Journal, 12: 163–178, (2016).
[17] Trenchea, C., “Second-order unconditionally stable IMEX schemes: Implicit for local effects and explicit for nonlocal effects”, ROMAI Journal, 12: 163–178, (2016).
[18] Schäfer, M., Turek, S., “Benchmark computations of laminar flow around a cylinder”, In: Hirschel E.H. (eds) Flow Simulation with High-Performance Computers II. Notes on Numerical Fluid Mechanics (NNFM), Vieweg+Teubner Verlag, 48: 547–566, (1996).
[18] Schäfer, M., Turek, S., “Benchmark computations of laminar flow around a cylinder”, In: Hirschel E.H. (eds) Flow Simulation with High-Performance Computers II. Notes on Numerical Fluid Mechanics (NNFM), Vieweg+Teubner Verlag, 48: 547–566, (1996).
[19] Heywood, J., Rannacher, R., “Finite element approximation of the nonstationary Navier-Stokes problem, Part II: Stability of solutions and error estimates uniform in time”, SIAM Journal on Numerical Analysis, 23: 750–777, (1986).
[19] Heywood, J., Rannacher, R., “Finite element approximation of the nonstationary Navier-Stokes problem, Part II: Stability of solutions and error estimates uniform in time”, SIAM Journal on Numerical Analysis, 23: 750–777, (1986).
[20] Mohebujjaman, M., Rebholz, L., Xie, X., Iliescu, T., “Energy balance and mass conservation in reduced order models of fluid flows”, Journal of Computational Physics, 346: 262–277, (2017).
[20] Mohebujjaman, M., Rebholz, L., Xie, X., Iliescu, T., “Energy balance and mass conservation in reduced order models of fluid flows”, Journal of Computational Physics, 346: 262–277, (2017).