On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems

Boyce E. Griffith, Charles Peskin

Research output: Contribution to journalArticle

Abstract

The immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a (visco-)elastic structure. In [M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998; M.-C. Lai, C.S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 705-719], Lai and Peskin introduced a formally second order accurate immersed boundary method, but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions. Consequently, in practice only first order convergence rates have been observed. In the present work, we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers. We consider two sets of material properties for the viscoelastic structure, including a case where the material properties of the coupled system are discontinuous at the fluid-structure interface. For both sets of material properties, the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations.

Original languageEnglish (US)
Pages (from-to)75-105
Number of pages31
JournalJournal of Computational Physics
Volume208
Issue number1
DOIs
StatePublished - Sep 1 2005

Fingerprint

Materials properties
Fluids
Fluid structure interaction
Circular cylinders
Reynolds number
Viscosity
theses
fluids
incompressible fluids
circular cylinders
interactions
viscosity
formulations
simulation

Keywords

  • Convergence
  • Fluid-structure interaction
  • Immersed boundary method
  • Incompressible flow
  • Viscoelasticity

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

On the order of accuracy of the immersed boundary method : Higher order convergence rates for sufficiently smooth problems. / Griffith, Boyce E.; Peskin, Charles.

In: Journal of Computational Physics, Vol. 208, No. 1, 01.09.2005, p. 75-105.

Research output: Contribution to journalArticle

@article{f9689d81719849b8b3d5e3dae3fcf244,
title = "On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems",
abstract = "The immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a (visco-)elastic structure. In [M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998; M.-C. Lai, C.S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 705-719], Lai and Peskin introduced a formally second order accurate immersed boundary method, but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions. Consequently, in practice only first order convergence rates have been observed. In the present work, we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers. We consider two sets of material properties for the viscoelastic structure, including a case where the material properties of the coupled system are discontinuous at the fluid-structure interface. For both sets of material properties, the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations.",
keywords = "Convergence, Fluid-structure interaction, Immersed boundary method, Incompressible flow, Viscoelasticity",
author = "Griffith, {Boyce E.} and Charles Peskin",
year = "2005",
month = "9",
day = "1",
doi = "10.1016/j.jcp.2005.02.011",
language = "English (US)",
volume = "208",
pages = "75--105",
journal = "Journal of Computational Physics",
issn = "0021-9991",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - On the order of accuracy of the immersed boundary method

T2 - Higher order convergence rates for sufficiently smooth problems

AU - Griffith, Boyce E.

AU - Peskin, Charles

PY - 2005/9/1

Y1 - 2005/9/1

N2 - The immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a (visco-)elastic structure. In [M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998; M.-C. Lai, C.S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 705-719], Lai and Peskin introduced a formally second order accurate immersed boundary method, but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions. Consequently, in practice only first order convergence rates have been observed. In the present work, we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers. We consider two sets of material properties for the viscoelastic structure, including a case where the material properties of the coupled system are discontinuous at the fluid-structure interface. For both sets of material properties, the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations.

AB - The immersed boundary method is both a mathematical formulation and a numerical scheme for problems involving the interaction of a viscous incompressible fluid and a (visco-)elastic structure. In [M.-C. Lai, Simulations of the flow past an array of circular cylinders as a test of the immersed boundary method, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, 1998; M.-C. Lai, C.S. Peskin, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160 (2000) 705-719], Lai and Peskin introduced a formally second order accurate immersed boundary method, but the convergence properties of their algorithm have only been examined computationally for problems with nonsmooth solutions. Consequently, in practice only first order convergence rates have been observed. In the present work, we describe a new formally second order accurate immersed boundary method and demonstrate its performance for a prototypical fluid-structure interaction problem, involving an immersed viscoelastic shell of finite thickness, studied over a broad range of Reynolds numbers. We consider two sets of material properties for the viscoelastic structure, including a case where the material properties of the coupled system are discontinuous at the fluid-structure interface. For both sets of material properties, the true solutions appear to possess sufficient smoothness for the method to converge at a second order rate for fully resolved computations.

KW - Convergence

KW - Fluid-structure interaction

KW - Immersed boundary method

KW - Incompressible flow

KW - Viscoelasticity

UR - http://www.scopus.com/inward/record.url?scp=19044361846&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19044361846&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2005.02.011

DO - 10.1016/j.jcp.2005.02.011

M3 - Article

AN - SCOPUS:19044361846

VL - 208

SP - 75

EP - 105

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -