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Volume 17, Issue 1
Analysis of a Second-Order Decoupled Time-Stepping Scheme for Transient Viscoelastic Flow

S. S. Ravindran

Int. J. Numer. Anal. Mod., 17 (2020), pp. 87-109.

Published online: 2020-02

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  • Abstract

In this paper, we propose and analyze a decoupled second order backward difference formula (BDF2) time-stepping algorithm for solving transient viscoelastic fluid flow. The spatial discretization is based on continuous Galerkin finite element approximation for the velocity and pressure, and discontinuous Galerkin finite element approximation for the viscoelastic stress tensor. To obtain a non-iterative decoupled algorithm from the fully discrete nonlinear system, we employ a second order extrapolation in time to the nonlinear terms. The algorithm requires the solution of one Navier-Stokes problem and one constitutive equation per time step. For mesh size $h$ and temporal step size ∆$t$ sufficiently small satisfying ∆$t$ ≤ $Ch$$d$/4, a priori error estimates in terms of ∆$t$ and $h$ are derived. Numerical tests are presented that illustrate the accuracy and stability of the algorithm.

  • Keywords

Viscoelasticity, finite element method, discontinuous Galerkin method, decoupled scheme, error estimates, BDF2.

  • AMS Subject Headings

65N30, 65N12, 76A10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

ravinds@uah.edu (S. S. Ravindran)

  • BibTex
  • RIS
  • TXT
@Article{IJNAM-17-87, author = {Ravindran , S. S.}, title = {Analysis of a Second-Order Decoupled Time-Stepping Scheme for Transient Viscoelastic Flow}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2020}, volume = {17}, number = {1}, pages = {87--109}, abstract = {

In this paper, we propose and analyze a decoupled second order backward difference formula (BDF2) time-stepping algorithm for solving transient viscoelastic fluid flow. The spatial discretization is based on continuous Galerkin finite element approximation for the velocity and pressure, and discontinuous Galerkin finite element approximation for the viscoelastic stress tensor. To obtain a non-iterative decoupled algorithm from the fully discrete nonlinear system, we employ a second order extrapolation in time to the nonlinear terms. The algorithm requires the solution of one Navier-Stokes problem and one constitutive equation per time step. For mesh size $h$ and temporal step size ∆$t$ sufficiently small satisfying ∆$t$ ≤ $Ch$$d$/4, a priori error estimates in terms of ∆$t$ and $h$ are derived. Numerical tests are presented that illustrate the accuracy and stability of the algorithm.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13642.html} }
TY - JOUR T1 - Analysis of a Second-Order Decoupled Time-Stepping Scheme for Transient Viscoelastic Flow AU - Ravindran , S. S. JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 87 EP - 109 PY - 2020 DA - 2020/02 SN - 17 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13642.html KW - Viscoelasticity, finite element method, discontinuous Galerkin method, decoupled scheme, error estimates, BDF2. AB -

In this paper, we propose and analyze a decoupled second order backward difference formula (BDF2) time-stepping algorithm for solving transient viscoelastic fluid flow. The spatial discretization is based on continuous Galerkin finite element approximation for the velocity and pressure, and discontinuous Galerkin finite element approximation for the viscoelastic stress tensor. To obtain a non-iterative decoupled algorithm from the fully discrete nonlinear system, we employ a second order extrapolation in time to the nonlinear terms. The algorithm requires the solution of one Navier-Stokes problem and one constitutive equation per time step. For mesh size $h$ and temporal step size ∆$t$ sufficiently small satisfying ∆$t$ ≤ $Ch$$d$/4, a priori error estimates in terms of ∆$t$ and $h$ are derived. Numerical tests are presented that illustrate the accuracy and stability of the algorithm.

Ravindran , S. S.. (2020). Analysis of a Second-Order Decoupled Time-Stepping Scheme for Transient Viscoelastic Flow. International Journal of Numerical Analysis and Modeling. 17 (1). 87-109. doi:
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