Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

Stockie, John M.; Wetton, Brian

doi:10.1006/jcph.1999.6297

Cited by 105 publications

(73 citation statements)

References 22 publications

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“…The leading order term we derive above is calculated analytically using the spacecontinuous formulation with an unsmoothed Dirac delta function. As Stockie and Wetton pointed out in [32], this analysis over-predicts the stiffness of the Immersed Boundary method in a practical computation. If we use the leading order approximation directly, the semi-implicit scheme with the leading order terms derived above tends to over-dissipate the solution.…”

Section: The Numerical Schemementioning

confidence: 86%

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Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations

Hou¹,

Shi²

2008

Journal of Computational Physics

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The Immersed Boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semiimplicit schemes to remove this stiffness from the Immersed Boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the Small Scale Decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semiimplicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property.

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Section: The Numerical Schemementioning

confidence: 86%

“…One of the main difficulties that the Immersed Boundary method encounters is that it suffers from a severe timestep restriction in order to keep the stability [27,32,30]. This has been the major limitation of the Immersed Boundary method.…”

Section: Introductionmentioning

confidence: 99%

Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations

Hou¹,

Shi²

2008

Journal of Computational Physics

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“…One of the most commonly used schemes with an explicit treatment of the immersed boundary is the so-called Forward Euler/Backward Euler (FE/BE) [3] in which the tension force is explicit (Forward Euler) and the viscous term is implicit (Backward Euler). That is,…”

Section: Some Comments On Computational Costs and Efficiencymentioning

confidence: 99%

“…In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce severe time-step restrictions for explicit discretization [2,3]. Fully implicit discretizations and some suitable semi-implicit schemes remove this hindering constraint but seemingly at a cost that makes these options impractical [4,5].…”

Section: Introductionmentioning

confidence: 99%

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

Ceniceros

Fisher

Roma

2009

Journal of Computational Physics

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The Immersed Boundary Method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 70's which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged, leads to a linear system of equations for the interface configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin's lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system's matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve.

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“…As is well known, the IB method suffers a time step restriction to maintain numerical stability [21,27,29,30]. This restriction becomes more stringent when the elastic force is stiff and the force spreading occurs at the beginning of each time step (an explicit scheme).…”

Section: Introductionmentioning

confidence: 99%

An Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

Lai

2013

E Asian J Appl Math

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Abstract.We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.

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Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes

Cited by 105 publications

References 22 publications

Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations

Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

An Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

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