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

Thomas G. Fai, Boyce E. Griffith, Yoichiro Mori, Charles Peskin

Research output: Contribution to journalArticle

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 languageEnglish (US)
JournalSIAM Journal on Scientific Computing
Volume36
Issue number3
DOIs
StatePublished - 2014

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Variable Viscosity
Immersed Boundary Method
Viscosity
Gravity waves
Coefficient
Navier Stokes equations
Linear systems
Time Stepping
Fluids
First-order
Stratified Fluid
Internal Waves
Gravity Waves
Speed of Convergence
Stability and Convergence
Range of data
Navier-Stokes Equations
Linear Systems
Iteration
Predict

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

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.

In: SIAM Journal on Scientific Computing, Vol. 36, No. 3, 2014.

Research output: Contribution to journalArticle

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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.

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