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The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations

Aytekin Çibik, Farjana Siddiqua, William Layton

TL;DR

This work studies a Ramshaw–Mesina hybrid regularization for the incompressible Navier-Stokes equations using finite-element spatial discretization and implicit time stepping. The authors establish unconditional stability for the continuous model and a backward-Euler discretization, derive semi-discrete error bounds with an exponential growth factor, and provide extensive numerical validation against penalty and artificial-compression methods. The results show that the hybrid scheme effectively decouples velocity and pressure and damps divergence and pressure oscillations, though the damping behavior in implicit-time implementations can differ from prior explicit-time predictions. The study offers a robust, practical approach for solving the NSE with improved stability and controlled divergence, supported by comprehensive theory and benchmark-like numerical tests.

Abstract

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.

The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations

TL;DR

This work studies a Ramshaw–Mesina hybrid regularization for the incompressible Navier-Stokes equations using finite-element spatial discretization and implicit time stepping. The authors establish unconditional stability for the continuous model and a backward-Euler discretization, derive semi-discrete error bounds with an exponential growth factor, and provide extensive numerical validation against penalty and artificial-compression methods. The results show that the hybrid scheme effectively decouples velocity and pressure and damps divergence and pressure oscillations, though the damping behavior in implicit-time implementations can differ from prior explicit-time predictions. The study offers a robust, practical approach for solving the NSE with improved stability and controlled divergence, supported by comprehensive theory and benchmark-like numerical tests.

Abstract

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.
Paper Structure (15 sections, 6 theorems, 68 equations, 7 figures, 1 table)

This paper contains 15 sections, 6 theorems, 68 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation and Preliminaries
  4. Formulation
  5. Stability
  6. Error Analysis
  7. Numerical Tests
  8. Numerical Stability
  9. Test of Accuracy
  10. Damping of Oscillations
  11. Backward Euler
  12. Backward Euler + time filter
  13. Trapezoidal
  14. Flow over a full step inside a channel
  15. Conclusion and future prospects

Key Result

Lemma 2.3

\newlabeltrilinear_ineq0 (see cfdbook). There exists $C_1$ and $C_2$ such that for any $u,\ v,\ w\in X$, the skew-symmetric tri-linear form, $b^{*}(u,v,w)$ satisfies

Figures (7)

  • Figure 1: Velocity and pressure norms for different selections of $\alpha$ and $\beta$.
  • Figure 2: Triangulated computational domain.
  • Figure 3: Offset circles test results for BE time discretization.
  • Figure 4: Offset circles test results for BE+time filter.
  • Figure 5: Offset circles test results for trapezoidal discretization.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proof 1
  • Remark 2.5
  • Theorem 3.1
  • Remark 3.2
  • Theorem 4.1
  • Proof 2
  • ...and 6 more