23 research outputs found
Proper orthogonal decomposition closure models for fluid flows: Burgers equation
This paper puts forth several closure models for the proper orthogonal
decomposition (POD) reduced order modeling of fluid flows. These new closure
models, together with other standard closure models, are investigated in the
numerical simulation of the Burgers equation. This simplified setting
represents just the first step in the investigation of the new closure models.
It allows a thorough assessment of the performance of the new models, including
a parameter sensitivity study. Two challenging test problems displaying moving
shock waves are chosen in the numerical investigation. The closure models and a
standard Galerkin POD reduced order model are benchmarked against the fine
resolution numerical simulation. Both numerical accuracy and computational
efficiency are used to assess the performance of the models
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
Structure-preserving Reduced Order Modeling of non-traditional Shallow Water Equation
An energy preserving reduced order model is developed for the nontraditional
shallow water equation (NTSWE) with full Coriolis force. The NTSWE in the
noncanonical Hamiltonian/Poisson form is discretized in space by finite
differences. The resulting system of ordinary differential equations is
integrated in time by the energy preserving average vector field (AVF) method.
The Poisson structure of the NTSWE in discretized exhibits a skew-symmetric
matrix depending on the state variables. An energy preserving, computationally
efficient reduced-order model (ROM) is constructed by proper orthogonal
decomposition with Galerkin projection. The nonlinearities are computed for the
ROM efficiently by discrete empirical interpolation method. Preservation of the
semi-discrete energy and the enstrophy are shown for the full order model, and
for the ROM which ensures the long term stability of the solutions. The
accuracy and computational efficiency of the ROMs are shown by two numerical
test problemsComment: 17 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1907.0940
POD-DEIM Based Model Order Reduction for the Spherical Shallow Water Equations with Turkel-Zwas Finite Difference Discretization
We consider the shallow water equations (SWE) in spherical coordinates solved by Turkel-Zwas (T-Z) explicit large time-step scheme. To reduce the dimension of the SWE model, we use a well-known model order reduction method, a proper orthogonal decomposition (POD). As the computational complexity still depends on the number of variables of the full spherical SWE model, we use discrete empirical interpolation method (DEIM) proposed by Sorensen to reduce the computational complexity of the reduced-order model. DEIM is very helpful in evaluating quadratically nonlinear terms in the reduced-order model. The numerical results show that POD-DEIM is computationally very efficient for implementing model order reduction for spherical SWE
Structure Preserving Model Order Reduction of Shallow Water Equations
In this paper, we present two different approaches for constructing
reduced-order models (ROMs) for the two-dimensional shallow water equation
(SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of
the SWE. After integration in time by the fully implicit average vector field
method, ROMs are constructed with proper orthogonal decomposition/discrete
empirical interpolation method (POD/DEIM) that preserves the Hamiltonian
structure. In the second approach, the SWE as a partial differential equation
with quadratic nonlinearity is integrated in time by the linearly implicit
Kahan's method and ROMs are constructed with the tensorial POD that preserves
the linear-quadratic structure of the SWE. We show that in both approaches, the
invariants of the SWE such as the energy, enstrophy, mass, and circulation are
preserved over a long period of time, leading to stable solutions. We conclude
by demonstrating the accuracy and the computational efficiency of the reduced
solutions by a numerical test problem