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On conservation laws of Navier-Stokes Galerkin discretizations

Sergey Charnyi, Timo Heister, Maxim A. Olshanskii, Leo G. Rebholz

arXiv:1605.09763v2math.NA

TL;DR

Weak divergence enforcement in Galerkin Navier–Stokes discretizations can disrupt physical conservation laws. The paper introduces the EMA-conserving formulation, which preserves a broad set of quantities without strong divergence enforcement, and verifies these properties analytically and numerically.

  • Problem

    Galerkin discretizations with only weakly enforced divergence may fail to conserve energy, momentum, angular momentum, helicity, or vorticity.

  • Method

    The paper constructs the EMA-conserving formulation by choosing the nonlinear term to preserve discrete conservation laws and analyzes its convergence and vorticity-based quantities.

  • Results

    The EMA-conserving formulation conserves energy, momentum, angular momentum, suitable vorticity, helicity, and 2D enstrophy, while numerical experiments verify its properties and performance.

  • Takeaways & Limitations

    Conservation can be recovered across multiple physical quantities without strongly enforcing the divergence constraint.

  • Takeaways & Limitations

    The initial study solves the nonlinear problem at every timestep, whereas common linearized schemes require only one linear solve per timestep.

Abstract

from arXiv · show

We study conservation properties of Galerkin methods for the incompressible Navier-Stokes equations, without the divergence constraint strongly enforced. In typical discretizations such as the mixed finite element method, the conservation of mass is enforced only weakly, and this leads to discrete solutions which may not conserve energy, momentum, angular momentum, helicity, or vorticity, even though the physics of the Navier-Stokes equations dictate that they should. We aim in this work to construct discrete formulations that conserve as many physical laws as possible without utilizing a strong enforcement of the divergence constraint, and doing so leads us to a new formulation that conserves each of energy, momentum, angular momentum, enstrophy in 2D, helicity and vorticity (for reference, the usual convective formulation does not conserve most of these quantities). Several numerical experiments are performed, which verify the theory and test the new formulation.

1 Introduction

The paper examines how weak enforcement of incompressibility affects conservation laws in Galerkin discretizations of Navier–Stokes equations. It introduces a formulation designed to preserve more physical quantities and evaluates its conservation properties numerically.

  • Scope: The study targets Galerkin methods including finite elements, isogeometric analysis, and spectral elements for incompressible Navier–Stokes equations.The equations model flows such as water, oil, and air, for which numerical approximation is needed because analytical solutions are difficult.
  • Motivation: Weakly enforced mass conservation can produce discrete velocities with nonzero divergence and loss of energy, momentum, and angular momentum conservation.The divergence error may remain significant on practical meshes despite convergence bounds.
  • Contribution: The paper develops a nonlinear formulation that preserves energy, momentum, angular momentum, vorticity, helicity, and 2D enstrophy without strong divergence enforcement.The formulation is called the Energy, Momentum, and Angular Momentum conserving formulation.
  • Evaluation: The paper combines formal conservation analysis with numerical experiments testing conservation properties and accuracy across formulations.The experiments examine the proposed formulation alongside existing schemes.

2 Notation and preliminaries

The preliminaries establish identities for non-divergence-free velocity and vorticity fields and use them to parameterize alternative vorticity formulations. These identities motivate the construction of a formulation with improved discrete conservation properties.

  • Function-space setting: The analysis permits H1 velocity fields without assuming that the fields are solenoidal, matching the weakly enforced divergence setting.Divergence-free assumptions are reserved for specific later equations.
  • Preliminary identities: A trilinear form and integration-by-parts identities provide the algebraic tools for analyzing nonlinear terms when divergence is not pointwise zero.These identities are used to represent inertia terms in alternative forms.
  • Vector notation: The symmetric part of the velocity gradient and the two- and three-dimensional curl conventions support the subsequent vector identities.The 2D curl is defined through a three-dimensional extension with zero third component.
  • Parameterized vorticity equation: A parameterized vorticity equation generates a family of formulations equivalent for divergence-free velocity and vorticity but distinct in the discrete setting.The parameter choices determine which discrete conservation properties are recovered.
  • Parameterized vorticity equation: The helical-density variable has physical meaning when β2 is nonzero, whereas for β2 = 0 it acts as a Lagrange multiplier for the divergence constraint.The paper selects parameters so the discrete vorticity equation delivers desired conserved quantities.

3 Conservation properties under div u̸ = 0 and the EMA formulation for Navier-Stokes

The paper analyzes conservation under weak divergence enforcement and introduces the EMA-conserving nonlinear term. Its central result is that this formulation uniquely preserves energy, momentum, and angular momentum among the compared forms, while also supporting additional conserved quantities.

  • Weak divergence enforcement: Weak divergence enforcement can leave practical divergence errors large, while enlarging the pressure space may violate inf-sup stability.This creates a setting where conservation cannot be assumed from pointwise incompressibility.
  • EMA formulation: The EMA-conserving formulation uses NL_emac(u) = 2D(u)u + (div u)u and modifies pressure by a velocity contribution.The formulation is based on an identity for the inertia term.
  • Conservation result: Among the five formulations, only EMA conserves energy, momentum, and angular momentum when divergence is weakly enforced.The result is stated for the corresponding conservation settings and suitable forcing assumptions.
  • Additional conservation laws: The proposed formulation also conserves suitably defined helicity, vorticity, and two-dimensional enstrophy.These quantities are defined through the discrete vorticity construction rather than requiring pointwise divergence-free velocity.

3.1 Energy, momentum and angular momentum

The conservation analysis shows that the EMA-conserving formulation preserves kinetic energy, momentum, and angular momentum under weak divergence enforcement. Other formulations preserve only subsets of these quantities or fail to conserve them in general.

  • Main result: Only the EMA-conserving formulation conserves all three quantities—kinetic energy, momentum, and angular momentum—under weak divergence enforcement.Skew-symmetric and rotational forms conserve kinetic energy, while EMA and conservative forms conserve momentum and angular momentum under the stated forcing assumptions.
  • Energy: For EMA, the nonlinear energy term vanishes, yielding kinetic-energy conservation when viscosity and forcing are absent.The identity is (NL_emac(u), u) = 0.
  • Energy: The convective and conservative formulations do not generally conserve kinetic energy when the discrete velocity is not divergence-free.The convective nonlinear term does not vanish in general, and the conservative form has the same limitation for energy.
  • Momentum: The conservative formulation conserves momentum, whereas the convective and rotational forms retain divergence-dependent nonlinear contributions.Forcing must have zero linear momentum for the conservation statement.
  • Angular momentum: EMA conserves angular momentum because its nonlinear contribution against the angular-momentum test functions vanishes.The test functions are φ_i = x × e_i, with zero divergence and Laplacian.

3.2 Vorticity, helicity and 2D enstrophy

The EMA formulation uses finite element vorticity equations to define discrete helicity, vorticity, and 2D enstrophy, which it conserves without strong divergence enforcement.

  • 3.2 Vorticity, helicity and 2D enstrophy: The discrete vorticity equation is parameterized by β1, β2, β3, and β4 to select formulations with desired conservation properties.These formulations are equivalent when velocity and vorticity are divergence-free but differ in their discrete behavior.
  • 3.2 Vorticity, helicity and 2D enstrophy: EMA conserves helicity, 2D enstrophy, and total vorticity through suitably defined finite element vorticity solutions.The preserved quantities are H=(u,w1), H2D, and the components of total vorticity.
  • 3.2 Vorticity, helicity and 2D enstrophy: For total vorticity, testing with constant basis functions and using their divergence-free property makes the nonlinear contributions cancel.The resulting conserved components are Wi for i=1,2,3.
  • 3.2 Vorticity, helicity and 2D enstrophy: In 2D, choosing the reduced vorticity formulation and testing with w establishes conservation of the discrete enstrophy H2D.The stated condition is curl f=0 and ν=0.
  • 3.2 Vorticity, helicity and 2D enstrophy: The helicity proof couples the velocity equation with a companion vorticity equation and uses a vector identity to show that (u,w1) is conserved.The analysis assumes zero forcing and viscosity.

3.3 Conservation properties of EMA-conserving formulation with Crank-Nicolson timestepping

The Crank–Nicolson EMA scheme preserves the paper’s discrete conservation laws under the stated assumptions, extending the semi-discrete results to fully discrete time stepping.

  • 3.3 Conservation properties of EMA-conserving formulation with Crank-Nicolson timestepping: Crank–Nicolson EMA exactly conserves kinetic energy, momentum, angular momentum, helicity, vorticity, and 2D enstrophy without forcing or viscosity.This is stated as Proposition 3.3 for the fully discrete scheme.
  • 3.3 Conservation properties of EMA-conserving formulation with Crank-Nicolson timestepping: With ν=0 and f=0, the Crank–Nicolson energy balance reduces to exact energy conservation.The time-discrete balance uses the composite midpoint approximation in time.
  • 3.3 Conservation properties of EMA-conserving formulation with Crank-Nicolson timestepping: Under zero-momentum forcing and the stated interior-support assumption, the scheme exactly conserves each component of momentum.The proof uses constant test functions through the restriction χ(ei).
  • 3.3 Conservation properties of EMA-conserving formulation with Crank-Nicolson timestepping: The fully discrete helicity and vorticity analyses introduce companion vorticity solutions whose existence, rather than computation, suffices for the conservation proofs.The same construction also supports the 2D enstrophy result.
  • 3.3 Conservation properties of EMA-conserving formulation with Crank-Nicolson timestepping: The discrete vorticity components satisfy (w^{n+1},ei)=(w^n,ei), yielding exact total-vorticity conservation.The nonlinear terms vanish under the constant test functions and the stated assumptions.

3.4 Convergence of the EMA-conserving scheme

The EMA-conserving scheme retains the convergence analysis of the skew-symmetric formulation because its nonlinearity satisfies the same energy-orthogonality property.

  • 3.4 Convergence of the EMA-conserving scheme: The skew-symmetric convergence proofs extend to EMA for both semi-discrete and Crank–Nicolson fully discrete schemes.The extension applies to the stated convergence analysis for the true Navier–Stokes solution.
  • 3.4 Convergence of the EMA-conserving scheme: The key property is (NLemac(u),u)=0, which allows the existing skew-symmetric error analysis to be adapted.The passage states that proofs for other skew-symmetric time-stepping methods can also be adapted.

3.5 Discussion

The EMA formulation is uniquely identified among the considered inertia-term combinations by simultaneous conservation of discrete kinetic energy, momentum, and angular momentum.

  • 3.5 Discussion: The EMA formulation is the unique combination found to conserve discrete kinetic energy, momentum, and angular momentum.Helicity, 2D enstrophy, and vorticity are understood using companion finite element vorticity equations.
  • 3.5 Discussion: The companion vorticity equation is introduced only to define suitable discrete conserved quantities and is not part of the finite element method.The paper does not claim that EMA is the only or simplest formulation conserving all listed quantities.

4 Numerical Experiments

Numerical experiments compare several Navier–Stokes formulations across idealized and realistic flows. EMAC preserves the broadest set of conservation properties and generally provides stable, accurate results, although other formulations can perform comparably in some benchmarks.

  • 4 Numerical Experiments: The experiments use Taylor-Hood elements with multiple time integrators and test conservation, accuracy, and stability across idealized and viscous flow problems.The Gresho problem uses Crank–Nicolson, while other tests include BDF3 and varied meshes and boundary settings.
  • 4.1 Gresho Problem: EMAC conserves energy, momentum, and angular momentum in the Gresho problem, while also achieving the best L2(Ω) velocity accuracy.CONV and CONS experience energy blowup; ROT and SKEW remain stable but are less accurate.
  • 4.2 Channel flow around a cylinder: The cylinder benchmark shows comparable formulations overall, with EMAC best for maximum drag, CONV and SKEW best for maximum lift, and EMAC, CONV, and CONS best for pressure drop.These comparisons use maximum lift and drag coefficients and pressure drop at t=8.
  • 4.3 Channel flow past a flat plate at Re=100: For flat-plate flow, EMAC’s average drag most closely matches the very fine discretization, while CONV, EMAC, and SKEW have similar recirculation-point accuracy.ROT and CONS fail to complete because of instability and energy growth.
  • 4.4 Channel flow past a forward-backward facing step: In the forward-backward step test, CONS fails on both meshes, whereas EMAC, CONV, SKEW, and ROT remain stable to T=60 with differing accuracy.On the finer mesh, EMAC, CONV, and SKEW match literature solutions, while ROT is poor; on the coarse mesh, CONV is best.
  • 4.5 Driven Cavity for Re=10,000: The driven-cavity results show mesh convergence and good agreement with reference statistics using a timestep of 0.02 and a mesh with 592,387 degrees of freedom.The computed statistics agree with prior results despite using a much coarser mesh than that study.

5 Conclusions and Future Directions

The EMAC formulation conserves a broad set of quantities under weak solenoidal enforcement, and numerical experiments verify its conservation properties and comparative performance. Future work targets more efficient nonlinear treatments because the current schemes solve the nonlinear problem at every timestep.

  • EMAC conserves energy, momentum, angular momentum, appropriately defined vorticity, helicity, and 2D enstrophy despite weak enforcement of the solenoidal constraint.
  • Common convective, conservative, rotational, and skew-symmetric formulations do not each conserve energy, momentum, and angular momentum under weak divergence enforcement.
  • The numerical experiments verify EMAC’s conservation properties and show that it performs at least as well as, or better than, commonly used formulations.
  • The current EMAC treatment solves the nonlinear problem at every timestep, whereas common linearized schemes require only one linear solve per timestep.
  • Future work should develop more efficient treatments of EMAC, since Newton-based nonlinear solves often require two or three linear solves per timestep.
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