On-line and Off-line POD Assisted Projective Integral for Non-linear Problems: A Case Study with Burgers-Equation
The POD-assisted projective integration method based on the equation-free framework is presented in this paper. The method is essentially based on the slow manifold governing of given system. We have applied two variants which are the “on-line" and “off-line" methods for solving the one-dimensional viscous Bergers- equation. For the on-line method, we have computed the slow manifold by extracting the POD modes and used them on-the-fly along the projective integration process without assuming knowledge of the underlying slow manifold. In contrast, the underlying slow manifold must be computed prior to the projective integration process for the off-line method. The projective step is performed by the forward Euler method. Numerical experiments show that for the case of nonperiodic system, the on-line method is more efficient than the off-line method. Besides, the online approach is more realistic when apply the POD-assisted projective integration method to solve any systems. The critical value of the projective time step which directly limits the efficiency of both methods is also shown.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1328540Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 953
 N.M. Arifin, M.S.M Noorani, and A. Kilicman. Modelling of marangoni convection using proper orthogonal decomposition. Nonlinear Dynamics, 48(3):331-337, 2007.
 G. Bekooz, P. Holmes, and J.L. Lumley. The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech., 25:539-575, 1993.
 W. Cazemier, R.W. Verstappen, and A.E. Veldman. Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids, 10 (7):1685-1699, 1998.
 A.E. Deane, I.G. Kevrekidis, G.E. Karniadakis, and S.A. Orszag. Lowdimensional models for complex geometry flows: Application to grooved channels and circular cylinders. Phys. Fluids A, 3 (10):2337-2354, 1991.
 C.W. Gear and I.G. Kevrekidis. Telescopic projective methods for parabolic differential equations. J. Comput. Phys., 187(1):95-109, 2003.
 C.W. Gear and I.G. Kevrekidis. Constraint-defined manifolds: a legacy code approach to low-dimensional computation. J. Sci. Comput., 25(1):17-28, 2004.
 I.G. Kevrekidis, C.W. Gear, and G. Hummer. Equation-free: The computer-aided analysis of complex multiscale systems. AIChE Journal 50 (7), pp. 1346-1355, 50(7):1346-1355, 2004.
 I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P.G. Kevrekidis, O. Runborg, and C. Theodoropoulos. Equation-free coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Comm. Math. Sci., 1(4):715-762, 2003.
 X. Ma, G.S. Karamanos, and G.E. Karniadakis. Dynamics and lowdimensionality of the turbulent near-wake. J. Fluid Mech., 410:29-65, 2000.
 X. Ma and G.E. Karniadakis. A low-dimensional model for simulating 3d cylinder flow. J. Fluid Mech., 458:181-190, 2002.
 A. Makeev, D. Maroudas, and I.G. Kevrekidis. Coarse stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples. J. Chem. Phys., 116:10083-10091, 2002.
 A. Makeev, D. Maroudas, A. Panagiotopoulos, and I.G. Kevrekidis. Coarse bifurcation analysis of kinetic Monte Carlo simulations: a latticegas model with lateral interactions. J. Chem. Phys., 117:8229-8240, 2002.
 A. Missoffe, J. Juillard, and D. Aubry. Reduced-order modelling of the reynolds equation for flexible structures. In Technical Proceedings of NSTI Nanotech 2007, pages 137-140, 2007.
 H. Nguyen and J. Reynen. A space-time finite element approach to burgers equation. In Numerical Methods for Non-Linear Problems, Vol.2. Pineridge Publisher, Swansea, 1982.
 B.R. Noack, K. Afanasiev, M. Morzy'nski, G. Tadmor, and F. Thiele. A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech., 497:335-363, 2003.
 J. Rambo and Y. Joshi. Reduced-order modeling of turbulent forced convection with parametric conditions. International Journal of Heat and Mass Transfer, 50(3-4):539-551, 2007.
 S.S. Ravindran. A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. International Journal for Numerical Methods in Fluids, 34(5):425-448, 2000.
 D. Rempfer. Low-dimensional modeling and numerical simulation of transition in simple shear flows. Annual Review of Fluid Mechanics, 35:229-265, 2003.
 R. Rico-Martinez, C.W. Gear, and I.G. Kevrekidis. Coarse projective kMC integration: forward/reverse initial and boundary value problems. J. Comput. Phys., 196:474-489, 2004.
 L. Russo, C.I. Siettos, and I.G. Kevrekidis. Reduced computations for nematic-liquid crystals: A timestepper approach for systems with continuous symmetries. Journal of Non-Newtonian Fluid Mechanics, 146(1-3):51-58, 2007.
 S. Sirisup and G.E. Karniadakis. A spectral viscosity method for correcting the long-term behavior of POD models. J. Comput. Phys., 194(1):92-116, 2004.
 S. Sirisup, G.E. Karniadakis, D. Xiu, and I.G. Kevrekidis. Equationfree/ galerkin-free pod-assisted computation of incompressible flows. J. Comput. Phys., 207(2):568-587, 2005.
 L. Sirovich. Turbulence and the dynamics of coherent structures, Parts I, II and III. Quart. Appl. Math., XLV:561-590, 1987.
 M. Samimy et. al. Feedback control of subsonic cavity flows using reduced-order models. J. Fluid Mech., 579:315-346, 2007.
 D. Xiu, I.G. Kevrekidis, and R. Ghanem. An equation-free, multiscale approach to uncertainty quantification. Computing in Science and Engineering, 7(3):16-23, 2005.