The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

doi:10.1006/jcph.2001.6813

Journal of Computational Physics

Volume 171, Issue 2, 10 August 2001, Pages 822-842

https://doi.org/10.1006/jcph.2001.6813 Get rights and content

Abstract

Peskin's Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain. However, the Immersed Boundary Method is known to be first-order accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier–Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a second-order projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper.

References (41)

J.B. Bell et al.
A second-order projection method for the incompressible Navier–Stokes equations
J. Comput. Phys.
(1989)
D.L. Brown et al.
Accurate projection methods for the incompressible Navier–Stokes equations
J. Comput. Phys.
(2001)
Y.C. Chang et al.
A level set formulation of Eulerian interface capturing method for incompressible fluid flows
J. Comput. Phys.
(1996)
R. Cortez et al.
The blob projection method for immersed boundary problems
J. Comput. Phys.
(2000)
R.H. Dillon et al.
Modeling biofilm processes using the immersed boundary method
J. Comput. Phys.
(1996)
L.J. Fauci
Interaction of oscillating filaments—A computational study
J. Comput. Phys.
(1990)
A.L. Fogelson
A mathematical model and numerical method for studying platelet adhesion and aggregation during blood clotting
J. Comput. Phys.
(1984)
T. Hou et al.
A hybrid method for moving interface problems with application to the Hele-Shaw flow
J. Comput. Phys.
(1997)
J. Kim et al.
Application of a fractional-step method to incompressible Navier–Stokes equations
J. Comput. Phys.
(1985)
M. Lai et al.
An immersed boundary method with formal second-order accuracy and reduced numerical viscosity
J. Comput. Phys.
(2000)

M.-C. Lai et al.

A remark on jump conditions for the three-dimensional Navier–Stokes equations involving an immersed moving membrane

Applied Math. Lett.

(2001)

Z. Li

A note on immersed interface methods for three dimensional elliptic equations

Computers Math. Appl.

(1996)

Z. Li

Immersed interface method for moving interface problems

Numer. Algorithms

(1997)

Z. Li et al.

A numerical study of electro-migration voiding by evolving level set functions on a fixed cartesian grid

J. Comput. Phys.

(1999)

X. Liu et al.

A boundary condition capturing method for Poisson's equation on irregular domain

J. comput. Phys.

(2000)

S. Osher et al.

Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations

J. Comput. Phys.

(1988)

C.S. Peskin

Numerical analysis of blood flow in the heart

J. Comput. Phys.

(1977)

C.S. Peskin et al.

Modeling prosthetic heart valves for numerical analysis of blood flow in the heart

J. Comput. Phys.

(1980)

E.G. Puckett et al.

A high-order projection method for tracking fluid interfaces in variable density incompressible flows

J. Comput. Phys.

(1997)

D. Sulsky et al.

A numerical method for suspension flow

J. Comput. Phys.

(1991)

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Regular ArticleThe Immersed Interface Method for the Navier–Stokes Equations with Singular Forces

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The Immersed Interface Method for the Navier–Stokes Equations with Singular Forces