Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations

Atzberger, Paul J.; Kramer, Peter R.

doi:10.1016/j.matcom.2008.01.004

Cited by 7 publications

(9 citation statements)

References 30 publications

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“…Since H IB is linear in the microstructure forces, without loss of generality we can consider the case of only two microstructure degrees of freedom. We denote these as X [1] , X [2] and the displacement vector by z = X [2] − X [1] . In making comparisons with other hydrodynamic coupling tensors we find it helpful to make use of approximate symmetries satisfied by H IB .…”

Section: Effective Hydrodynamic Couplingmentioning

confidence: 99%

Stochastic Eulerian Lagrangian methods for fluid–structure interactions with thermal fluctuations

Atzberger

2011

Journal of Computational Physics

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214

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a b s t r a c tWe present approaches for the study of fluid-structure interactions subject to thermal fluctuations. A mixed mechanical description is utilized combining Eulerian and Lagrangian reference frames. We establish general conditions for operators coupling these descriptions. Stochastic driving fields for the formalism are derived using principles from statistical mechanics. The stochastic differential equations of the formalism are found to exhibit significant stiffness in some physical regimes. To cope with this issue, we derive reduced stochastic differential equations for several physical regimes. We also present stochastic numerical methods for each regime to approximate the fluid-structure dynamics and to generate efficiently the required stochastic driving fields. To validate the methodology in each regime, we perform analysis of the invariant probability distribution of the stochastic dynamics of the fluid-structure formalism. We compare this analysis with results from statistical mechanics. To further demonstrate the applicability of the methodology, we perform computational studies for spherical particles having translational and rotational degrees of freedom. We compare these studies with results from fluid mechanics. The presented approach provides for fluid-structure systems a set of rather general computational methods for treating consistently structure mechanics, hydrodynamic coupling, and thermal fluctuations.

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Section: Effective Hydrodynamic Couplingmentioning

confidence: 99%

Stochastic Eulerian Lagrangian methods for fluid–structure interactions with thermal fluctuations

Atzberger

2011

Journal of Computational Physics

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214

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“…Notice that when the time step is taken small, the temporal integrator presented in (3.61) has weak first order accuracy. A detailed error analysis is given in [36].…”

Section: Spatial Discretizationmentioning

confidence: 99%

Simulation of Osmotic Swelling by the Stochastic Immersed Boundary Method

Fai

Atzberger

et al. 2015

SIAM J. Sci. Comput.

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“…(36) for continuous time but representing the delta correlations through a 1/∆t scaling of the amplitude would only be reasonable if the time step ∆t is smaller than the relaxation time of all system variables. A more detailed analysis presented in ; Atzberger and Kramer (2006) provides a numerical representation of the thermal fluctuations which remains accurate even when the time step underresolves some or all of the fluid degrees of freedom. In essence, we exploit the linearity of the fluid equations to derive a stochastic exponential time stepping scheme (Hairer and Lubich, 2000;García-Archilla et al, 1999;Kassam and Trefethen, 2005).…”

Section: Temporal Discretization Of Stochastic Immersed Boundary Equamentioning

confidence: 99%

“…As shown in Atzberger and Kramer (2006), this numerical method is accurate (in a stochastically strong sense) provided only that the time step ∆t is much smaller than the time scale over which the immersed structures move a distance comparable to the size a of the elementary particles which are used to discretize the structures. That is, the time step must resolve the structural degrees of freedom but need not resolve all (or even any) of the fluid modes.…”

Section: Temporal Discretization Of Stochastic Immersed Boundary Equamentioning

confidence: 99%

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On the foundations of the stochastic immersed boundary method

Kramer

Peskin

Atzberger

2008

Computer Methods in Applied Mechanics and Engineering

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We explore the theoretical foundations for the inclusion of thermal fluctuations in the immersed boundary method for simulating microscale fluid systems with immersed flexible structures, as in cellular and subcellular biology. We investigate in particular the physical validity of the thermal forcing scheme with respect to the coupling of fluid and immersed structural degrees of freedom and non-equilibrium conditions. We discuss also the shortcomings of a natural alternative scheme in which the thermal fluctuations are applied directly to the structural degrees of freedom through Langevin-type dynamics.

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Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations

Cited by 7 publications

References 30 publications

Stochastic Eulerian Lagrangian methods for fluid–structure interactions with thermal fluctuations

Stochastic Eulerian Lagrangian methods for fluid–structure interactions with thermal fluctuations

Simulation of Osmotic Swelling by the Stochastic Immersed Boundary Method

On the foundations of the stochastic immersed boundary method

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