Results 1 
7 of
7
Nonequilibrium entropy limiters in lattice Boltzmann methods
, 2008
"... We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite vo ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
We construct a system of nonequilibrium entropy limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious oscillations without blurring of shocks, and do not affect smooth solutions. In general, they do the same work for LBM as flux limiters do for finite differences, finite volumes and finite elements methods, but for LBM the main idea behind the construction of nonequilibrium entropy limiter schemes is to transform a field of a scalar quantity — nonequilibrium entropy. There are two families of limiters: (i) based on restriction of nonequilibrium entropy (entropy “trimming”) and (ii) based on filtering of nonequilibrium entropy (entropy filtering). The physical properties of LBM provide some additional benefits: the control of entropy production and accurate estimation of introduced artificial dissipation are possible. The constructed limiters are tested on classical numerical examples: 1D athermal shock tubes with an initial density ratio 1:2 and the 2D liddriven cavity for Reynolds numbers Re between 2000 and 7500 on a coarse 100 × 100 grid. All limiter constructions are applicable both for entropic and for nonentropic equilibria.
Entropy Balance and Dispersive Oscillations in Lattice Boltzmann Models
, 2009
"... We conduct an investigation into the dispersive postshock oscillations in the entropic latticeBoltzmann method (ELBM). To this end we use a root finding algorithm to implement the ELBM which displays fast cubic convergence and guaranties the proper sign of dissipation. The resulting simulation on ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
(Show Context)
We conduct an investigation into the dispersive postshock oscillations in the entropic latticeBoltzmann method (ELBM). To this end we use a root finding algorithm to implement the ELBM which displays fast cubic convergence and guaranties the proper sign of dissipation. The resulting simulation on the onedimensional shock tube shows no benefit in terms of regularization from using the ELBM over the standard LBGK method. We also conduct an experiment investigating of the LBGK method using median filtering at a single point per time step. Here we observe that significant regularization can be achieved.
Noncanonic observers for canonic models of neural oscillators
, 2009
"... We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We consider the problem of state and parameter estimation for a class of nonlinear oscillators defined as a system of coupled nonlinear ordinary differential equations. Observable variables are limited to a few components of state vector and an input signal. This class of systems describes a set of canonic models governing the dynamics of evoked potential in neural membranes, including HodgkinHuxley, HindmarshRose, FitzHughNagumo, and MorrisLecar models. We consider the problem of state and parameter reconstruction for these models within the classical framework of observer design. This framework offers computationallyefficient solutions to the problem of state and parameter reconstruction of a system of nonlinear differential equations, provided that these equations are in the socalled adaptive observer canonic form. We show that despite typical neural oscillators being locally observable they are not in the adaptive canonic observer form. Furthermore, we show that no parameterindependent diffeomorphism exists such that the original equations of these models can be transformed into the adaptive canonic observer form. We demonstrate, however, that for the class of HindmarshRose and FitzHughNagumo models, parameterdependent coordinate transformations can be used to render these systems into the adaptive observer canonical form. This allows reconstruction, at least partially and up to a (bi)linear transformation, of unknown state and parameter values with exponential rate of convergence. In order to avoid the problem of only partial reconstruction
Stable simulation of fluid flow with highReynolds
, 2007
"... number using Ehrenfests ’ steps ..."
(Show Context)
Stability and stabilisation of the lattice Boltzmann method: Magic steps and salvation operations
, 2008
"... The lattice Boltzmann method (LBM) is known to have stability deficiencies. For example, local blowups and spurious oscillations are readily observed when the method is used to model highReynolds fluid flow. Beginning from thermodynamic considerations, the LBM can be recognised as a discrete dynam ..."
Abstract
 Add to MetaCart
(Show Context)
The lattice Boltzmann method (LBM) is known to have stability deficiencies. For example, local blowups and spurious oscillations are readily observed when the method is used to model highReynolds fluid flow. Beginning from thermodynamic considerations, the LBM can be recognised as a discrete dynamical system generated by entropic involution and freeflight and the stability analysis is more natural. In this paper we solve the stability problem of the LBM on the basis of this thermodynamic point of view. The main instability mechanisms are identified. The simplest and most effective receipt for stabilisation adds no artificial dissipation, preserves the secondorder accuracy of the method, and prescribes coupled steps: to start from a local equilibrium, then, after freeflight, perform the overrelaxation collision, and after a second freeflight step go to new local equilibrium. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup. Demonstration of the proposed stable LBMs are provided by the numerical simulation of a 1D shock tube and the unsteady 2Dflow around a squarecylinder up to Reynolds number O(10000). 1