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.
[1] 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.
[2] 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.
[3] 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.
[4] 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.
[5] C.W. Gear and I.G. Kevrekidis. Telescopic projective methods for
parabolic differential equations. J. Comput. Phys., 187(1):95-109, 2003.
[6] 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.
[7] 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.
[8] 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.
[9] X. Ma, G.S. Karamanos, and G.E. Karniadakis. Dynamics and lowdimensionality
of the turbulent near-wake. J. Fluid Mech., 410:29-65,
2000.
[10] X. Ma and G.E. Karniadakis. A low-dimensional model for simulating
3d cylinder flow. J. Fluid Mech., 458:181-190, 2002.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] D. Rempfer. Low-dimensional modeling and numerical simulation of
transition in simple shear flows. Annual Review of Fluid Mechanics,
35:229-265, 2003.
[19] 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.
[20] 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.
[21] 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.
[22] 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.
[23] L. Sirovich. Turbulence and the dynamics of coherent structures, Parts
I, II and III. Quart. Appl. Math., XLV:561-590, 1987.
[24] M. Samimy et. al. Feedback control of subsonic cavity flows using
reduced-order models. J. Fluid Mech., 579:315-346, 2007.
[25] 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.
[1] 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.
[2] 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.
[3] 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.
[4] 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.
[5] C.W. Gear and I.G. Kevrekidis. Telescopic projective methods for
parabolic differential equations. J. Comput. Phys., 187(1):95-109, 2003.
[6] 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.
[7] 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.
[8] 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.
[9] X. Ma, G.S. Karamanos, and G.E. Karniadakis. Dynamics and lowdimensionality
of the turbulent near-wake. J. Fluid Mech., 410:29-65,
2000.
[10] X. Ma and G.E. Karniadakis. A low-dimensional model for simulating
3d cylinder flow. J. Fluid Mech., 458:181-190, 2002.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] D. Rempfer. Low-dimensional modeling and numerical simulation of
transition in simple shear flows. Annual Review of Fluid Mechanics,
35:229-265, 2003.
[19] 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.
[20] 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.
[21] 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.
[22] 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.
[23] L. Sirovich. Turbulence and the dynamics of coherent structures, Parts
I, II and III. Quart. Appl. Math., XLV:561-590, 1987.
[24] M. Samimy et. al. Feedback control of subsonic cavity flows using
reduced-order models. J. Fluid Mech., 579:315-346, 2007.
[25] 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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:49639", author = "Montri Maleewong and Sirod Sirisup", title = "On-line and Off-line POD Assisted Projective Integral for Non-linear Problems: A Case Study with Burgers-Equation", abstract = "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.
", keywords = "Projective integration, POD method, equation-free.", volume = "5", number = "7", pages = "924-9", }
