The functional equation truncation method for approximating slow invariant manifolds: A rapid method for computing intrinsic low-dimensional manifolds

Roussel, Marc R.; Tang, Terry

doi:10.1063/1.2402172

Cited by 12 publications

(10 citation statements)

References 33 publications

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“…As mentioned in the introduction, Maas and Pope introduced the widely used IDLM method in 1992 [28] where a local time scale analysis is performed via matrix decomposition of the Jacobian of the right hand side of the ODE system. In [39,40], Roussel could demonstrate the coincidence between the ILDM method and his FET approach. The operation concept of FET is shown for a planar system…”

Section: Intrinsic Low Dimensional Manifold (Ildm)/functional Equatiomentioning

confidence: 99%

“…Thus, we now have two equations (( 20) and ( 22)) in the two unknowns z 2 (t), z 2 (t) for every z 1 (t) allowing the computation of an approximation to the one-dimensional manifold by using an iterative method to solve (20). The resulting manifold is called Functional Equation Truncation Approximated (FETA) manifold [39,40]. Its approximation of the one-dimensional SIM is valid insofar as z 2 (t) is small.…”

Section: Intrinsic Low Dimensional Manifold (Ildm)/functional Equatio...mentioning

confidence: 99%

“…The use of those manifolds for the numerical solution of differential equations is obvious in the G-scheme [45]. Furthermore, there are other approaches, whereof a few are presented in Chapter 3 of this work, including the Invariant Constrained Equilibrium Edge PreImage Curve (ICE-PIC) introduced by Ren et al in 2006 [36,37], an iterative model reduction method called Zero-Derivative Principle (ZDP) presented in [12,46], the Flow Curvature Method proposed by Ginoux in [14], the Functional Equation Truncation (FET) approach by Roussel [39,40], and methods by Adrover et al [1,2] and Al-Khateeb et al [4]. Additionally, there is a trajectory-based optimization approach based on a variational principle proposed by Lebiedz et al, described in [22][23][24][25][26][27]35].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

Lebiedz¹,

Unger²

2016

Mathematical and Computer Modelling of Dynamical Systems

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Chemical kinetic models in terms of ordinary differential equations correspond to finite dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow mode-description of the full system compute approximations of low-dimensional attracting slow invariant manifolds and parameterize these manifolds in terms of a subset of chosen chemical species, the reaction progress variables. The invariance property suggests a slow invariant manifold to be constructed as (a bundle of) solution trajectories of suitable ordinary differential equation initial or boundary value problems. The focus of this work is on a discussion of fundamental and unifying geometric and analytical issues of various approaches to trajectory-based numerical approximation techniques of slow invariant manifolds that are in practical use for model reduction in chemical kinetics. Two basic concepts are pointed out reducing various model reduction approaches to a common denominator. In particular, we discuss our recent trajectory optimization approach in the light of these two concepts. We relate both of them in a variational boundary value viewpoint, propose a Hamiltonian formulation and conjecture its relation to conservation laws, (partial) integrability and symmetry issues as underlying fundamental principles and potentially unifying elements of diverse dimension reduction approaches.

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Section: Intrinsic Low Dimensional Manifold (Ildm)/functional Equatiomentioning

confidence: 99%

Section: Intrinsic Low Dimensional Manifold (Ildm)/functional Equatio...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

Lebiedz¹,

Unger²

2016

Mathematical and Computer Modelling of Dynamical Systems

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“…Due to its simplicity, the model continues to be the focus of numerous studies and a prototypical problem for model reduction. In the absence of an analytic solution, two approaches, the quasi-equilibrium approximation and the steady-state assumption, have been extensively used in the literature to find an expression for the rate of the catalytic step. , The Michaelis–Menten equilibrium analysis is valid if the substrate reaches equilibrium on a much faster timescale than the product is formed The geometric picture of the phase space evolution can be found in refs − , and a comprehensive study of the slow manifold for the Michaelis–Menten mechanism is presented in ref .…”

Section: Sqem Construction For Chemical Kineticsmentioning

confidence: 99%

Spectral Quasi-Equilibrium Manifold for Chemical Kinetics

et al. 2016

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The Spectral Quasi-Equilibrium Manifold (SQEM) method is a model reduction technique for chemical kinetics based on entropy maximization under constraints built by the slowest eigenvectors at equilibrium. The method is revisited here and discussed and validated through the Michaelis-Menten kinetic scheme, and the quality of the reduction is related to the temporal evolution and the gap between eigenvalues. SQEM is then applied to detailed reaction mechanisms for the homogeneous combustion of hydrogen, syngas, and methane mixtures with air in adiabatic constant pressure reactors. The system states computed using SQEM are compared with those obtained by direct integration of the detailed mechanism, and good agreement between the reduced and the detailed descriptions is demonstrated. The SQEM reduced model of hydrogen/air combustion is also compared with another similar technique, the Rate-Controlled Constrained-Equilibrium (RCCE). For the same number of representative variables, SQEM is found to provide a more accurate description.

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“…As is well-known, the standard procedure to obtain (17) is as follows. First we invert Equations (16) to express the state variables in terms of the multipliers, x eq = x eq (γ ∞ ); these relations are then inserted into (15) to express the values of the constraints as functions of the multipliers, c…”

Section: General Non-equilibrium Problemmentioning

confidence: 99%

The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics

Beretta

Keck

Janbozorgi

et al. 2012

Entropy

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The Rate-Controlled Constrained-Equilibrium (RCCE) method for the description of the time-dependent behavior of dynamical systems in non-equilibrium states is a general, effective, physically based method for model order reduction that was originally developed in the framework of thermodynamics and chemical kinetics. A generalized mathematical formulation is presented here that allows including nonlinear constraints in non-local equilibrium systems characterized by the existence of a non-increasing Lyapunov functional under the system's internal dynamics. The generalized formulation of RCCE enables to clarify the essentials of the method and the built-in general feature of thermodynamic consistency in the chemical kinetics context. In this paper, we work out the details of the method in a generalized mathematical-physics framework, but for definiteness we detail its well-known implementation in the traditional chemical kinetics framework. We detail proofs and spell out explicit functional dependences so as to bring out and clarify each underlying assumption of the method. In the standard context of chemical kinetics of ideal gas mixtures, we discuss the relations between the validity of the detailed balance condition off-equilibrium and the thermodynamic consistency of the method. We also discuss two examples of RCCE gas-phase combustion calculations to emphasize the constraint-dependent performance of the RCCE method.

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The functional equation truncation method for approximating slow invariant manifolds: A rapid method for computing intrinsic low-dimensional manifolds

Cited by 12 publications

References 33 publications

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models

Spectral Quasi-Equilibrium Manifold for Chemical Kinetics

The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics

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