### Abstract

We analyze the stability and convergence of first-order accurate and second-order accurate timestepping schemes for the Navier-Stokes equations with variable viscosity. These schemes are characterized by a mixed implicit/explicit treatment of the viscous term, in which a numerical parameter, determines the degree of splitting between the implicit and explicit contributions. The reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each timestep. Provided the parameter is within a permissible range, we prove that the first-order accurate and second-order accurate schemes are convergent. We show further that the efficiency of the second-order accurate scheme depends on how is chosen within the permissible range, and we discuss choices that work well in practice. We use parameters motivated by this analysis to simulate internal gravity waves, which arise in stratified fluids with variable density. We examine how the wave properties change in the nonlinear and variable viscosity regime, and we test how well our theory predicts the speed of convergence of the iteration used in the second-order accurate timestepping scheme.

Original language | English (US) |
---|---|

Journal | SIAM Journal on Scientific Computing |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- Convergence
- Immersed boundary method
- Incompressible flow
- Stability
- Variable viscosity

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*36*(3). https://doi.org/10.1137/12090304X

**Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers II : Theory.** / Fai, Thomas G.; Griffith, Boyce E.; Mori, Yoichiro; Peskin, Charles.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 36, no. 3. https://doi.org/10.1137/12090304X

}

TY - JOUR

T1 - Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers II

T2 - Theory

AU - Fai, Thomas G.

AU - Griffith, Boyce E.

AU - Mori, Yoichiro

AU - Peskin, Charles

PY - 2014

Y1 - 2014

N2 - We analyze the stability and convergence of first-order accurate and second-order accurate timestepping schemes for the Navier-Stokes equations with variable viscosity. These schemes are characterized by a mixed implicit/explicit treatment of the viscous term, in which a numerical parameter, determines the degree of splitting between the implicit and explicit contributions. The reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each timestep. Provided the parameter is within a permissible range, we prove that the first-order accurate and second-order accurate schemes are convergent. We show further that the efficiency of the second-order accurate scheme depends on how is chosen within the permissible range, and we discuss choices that work well in practice. We use parameters motivated by this analysis to simulate internal gravity waves, which arise in stratified fluids with variable density. We examine how the wave properties change in the nonlinear and variable viscosity regime, and we test how well our theory predicts the speed of convergence of the iteration used in the second-order accurate timestepping scheme.

AB - We analyze the stability and convergence of first-order accurate and second-order accurate timestepping schemes for the Navier-Stokes equations with variable viscosity. These schemes are characterized by a mixed implicit/explicit treatment of the viscous term, in which a numerical parameter, determines the degree of splitting between the implicit and explicit contributions. The reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each timestep. Provided the parameter is within a permissible range, we prove that the first-order accurate and second-order accurate schemes are convergent. We show further that the efficiency of the second-order accurate scheme depends on how is chosen within the permissible range, and we discuss choices that work well in practice. We use parameters motivated by this analysis to simulate internal gravity waves, which arise in stratified fluids with variable density. We examine how the wave properties change in the nonlinear and variable viscosity regime, and we test how well our theory predicts the speed of convergence of the iteration used in the second-order accurate timestepping scheme.

KW - Convergence

KW - Immersed boundary method

KW - Incompressible flow

KW - Stability

KW - Variable viscosity

UR - http://www.scopus.com/inward/record.url?scp=84901954180&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901954180&partnerID=8YFLogxK

U2 - 10.1137/12090304X

DO - 10.1137/12090304X

M3 - Article

AN - SCOPUS:84901954180

VL - 36

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 3

ER -