Vakonomic Fluids

Albert Chern; Anthony Gruber; Mohammad Sina Nabizadeh; Oliver Gross; Ritoban Roy-Chowdhury

REVIEW 3 major objections 6 minor 85 references

Vakonomic Fluids

T0 review · 3 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Forcing every variation to respect the discretized velocity constraint yields fluid paths that are true geodesics, conserving energy, enstrophy-like invariants, and circulation to machine precision.

desk verdict Clean vakonomic Lax derivation and a strong discrete Kelvin theorem, but the abstract oversells conservation and the continuum limit is unproven. read the letter →

arxiv 2607.18312 v2 pith:TIR6SILI submitted 2026-07-17 math-ph cs.GRcs.NAmath.DGmath.DSmath.MPmath.NAphysics.flu-dyn

Ritoban Roy-Chowdhury , Mohammad Sina Nabizadeh , Oliver Gross , Anthony Gruber , Albert Chern This is my paper · ORCID

classification math-phcs.GRcs.NAmath.DGmath.DSmath.MPmath.NAphysics.flu-dyn MSC 70F2576B0337K1053D2065M60

keywords vakonomicmechanicsincompressibleEulerequationsKoopmanrepresentationsub-RiemanniangeodesicsLaxequationLie–PoissonsystemsClebschvariablesKelvin'scirculationtheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Incompressible Euler flow is classically the geodesic equation on the group of volume-preserving diffeomorphisms; this paper asks what survives of that geodesic picture when the flow is discretized through the Koopman representation of diffeomorphisms as rotation matrices. That discretization imposes a nonholonomic constraint on admissible velocities, and the paper shows that enforcing it the vakonomic way — allowing only constraint-respecting variations — keeps the discrete flow a genuine geodesic on a sub-Riemannian matrix manifold. The stationarity condition closes as a Lax equation, dZ/dt = [Z^#, Z], which makes the system Lie–Poisson: kinetic energy, the entire spectrum (the Casimir invariants, discrete analogues of enstrophy and helicity), and a discrete Kelvin circulation theorem are all conserved to solver tolerance. A low-rank Clebsch momentum-map representation reduces the per-step cost from O(F^2) to O(mF) without sacrificing any of the structure, and experiments show stable, low-dispersion vortex evolution on coarse grids, on surfaces, and in 3D. The claim that would make this a full discretization theory — that the discrete dynamics converges to the Euler equations as the mesh refines — is stated and left open.

What carries the argument

The load-bearing object is the vakonomic variational principle applied to the right-invariant K-metric restricted to the distribution D_R = {XR : X ∈ im(A-bar)} in T SO(F). Lagrange–d'Alembert lets variations leave the distribution and therefore requires an ad hoc ambient-metric choice; the vakonomic principle instead confines the whole variation family to admissible paths, and the stationarity condition then closes as the Lax equation dZ/dt = [Z^#, Z]. The sharp operator # = A-bar K^-1 A-bar* is defined solely from the discrete advection operator and the velocity mass matrix K, so no metric information from outside the constraint space is needed — the heart of the paper's self-consistency c

What would settle it

Run the reset-free scheme on a Taylor–Green vortex at 32→64→128→256 resolution and measure the L^2 error of the vorticity field against the exact steady solution; if the error does not shrink at a stable order, the dynamics are structure-preserving but not a discretization of Euler. Alternatively, run a two-dipole leapfrogging experiment with resetting disabled: if tr(Z^2) and tr(Z^4) hold to machine precision while the vortex trajectories visibly miss the point-vortex prediction, then the exactly conserved structure is not the one that carries the fluid physics.

Watch

Extended reading notes

Core claim

Set up discrete fluid motion as a path R(t) in SO(F) whose body velocity X = dR/dt R^-1 is constrained to the image im(A-bar) of a discrete advection operator; because im(A-bar) is not closed under the matrix commutator, this is a genuine nonholonomic constraint. The paper proves that stationarity of the kinetic-energy action under variations that keep the family inside the constraint distribution — the vakonomic principle, as opposed to Lagrange–d'Alembert — is equivalent to a coadjoint evolution on so(F)* that reads, after the Frobenius identification, as the matrix Lax equation dZ/dt = [Z^#, Z], with sharp map # = A-bar K^-1 A-bar* built only from constraint-space data. This evolution is

Load-bearing premise

The whole construction rests on the premise that the constrained rotation-matrix dynamics actually approaches the incompressible Euler equations as the grid is refined — the paper proves exact structure preservation but explicitly leaves the convergence of its discrete Lie algebra to the divergence-free vector fields open; if that convergence fails, the method is a self-consistent toy rather than a discretization of Euler flow.

Editorial extensions

If this is right

Discrete ideal-fluid simulations become exactly isospectral: enstrophy- and helicity-like Casimirs no longer drift, removing a common source of spurious energy cascades in long runs.
One source of arbitrariness vanishes: unlike Lagrange–d'Alembert discretizations, the equations carry no free 'ambient metric' — only the choice of function space, velocity space, and mass matrix.
The low-rank Clebsch form reduces the per-step cost from O(F^2) to O(mF), making exactly structure-preserving ideal-fluid simulation practical on standard FEEC/B-spline and triangle-mesh grids without ever assembling a dense matrix.
The conserved discrete circulation is stronger than the weak circulation of earlier projected-Lax schemes: it holds for every loop transported by the flow, not merely in a projected sense.
The same variational framework extends to semidirect-product physics — stratified Boussinesq flow is demonstrated, and the authors sketch shallow-water, compressible, and MHD variants.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

My inference: the reset step, which the paper needs to curb dispersion, breaks the exact conservation the construction advertises; a natural testable extension is the paper's own suggestion of double-bracket dissipation, which would limit dispersion while keeping Casimirs exactly conserved.
My inference: the advertised clean separation of time-integration drift from resetting drift is only implicit in the figures; a reset-free diagnostic run that logs tr(Z^2) and tr(Z^4) every step would settle how much of the observed dissipation is actually the reset's doing.
My inference: the circulation theorem's physical content depends on how faithfully the Lie-algebraic closure of im(A-bar) represents real material loops; a coarse-mesh reset-free run against analytical point-vortex or shielded-vortex trajectories would quantify that fidelity.
My inference: the m = d choice of Clebsch pairs sets a fixed rank for Z, and the paper gives no rank-robustness test; sweeping m while monitoring the reconstructed velocity would show whether truncation in the low-rank ansatz is a source of error comparable to the discretization error.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline: this is a genuinely new variational route to discrete fluid dynamics, and the theory is cleaner than the abstract's marketing suggests—but the "machine-precision" conservation claim should be read as a reset-free statement, and whether this actually approximates incompressible Euler is still open.

What's new: the vakonomic treatment of the nonholonomic constraint that arises from Koopman discretization is a real departure from the standard Lagrange–d'Alembert approach. The derivation of the Lax equation (6.12) from the sub-Riemannian geodesic principle is rigorous and self-contained. The discrete Kelvin circulation theorem (Theorem 6.2) is stronger than the previous LdA result, and the low-rank Clebsch parameterization (Theorem 8.1) is a neat practical idea that preserves the Lax structure while cutting the cost from O(F^2) to O(mF). The numerical experiments, particularly the trefoil and leapfrogging cases, show qualitative improvements over baselines, and the paper is honest about its limitations—it explicitly flags the missing continuum limit and the resetting issue.

Where I'd push back: the missing continuum limit is not a minor technicality. The paper's central claim is that this is a discretization of the Euler equations, but Section 12 leaves open whether so(F) and im(A-bar) converge to sdiff(M). Without that, Eq. (6.12) is a self-consistent isospectral flow on a finite-dimensional space, but it's not established that it approximates Eq. (3.9). The Taylor–Green convergence test is evidence for one steady solution, not for the general initial-value problem. That said, this is not a fatal flaw for all readers—many structure-preserving schemes are used before convergence is proved—but it needs to be stated as an open question rather than an incidental remark.

The resetting issue is more concrete. The headline conservation of Casimirs holds only between resets; the actual algorithm resets frequently (every 0.21 s in most experiments), and each reset causes a jump. The energy drops about 1.6% over 5 s in one test, and the reset period is chosen per experiment by heuristics. The paper acknowledges this, but it means the abstract's "machine-precision satisfaction" is not what the production code delivers. That's a mismatch between the claim and the implementation.

The lack of released code is a minor concern for verifiability, but the paper describes the algorithm in enough detail that reimplementation is feasible.

Verdict: this is a serious paper with a real idea, and it deserves a serious referee. If I were the editor, I'd send it to peer review but ask the referees to weigh the continuum limit question and the resetting issue carefully. The authors should either prove convergence under reasonable assumptions or soften the claim to "a structure-preserving discrete model inspired by Euler," and they should present the reset-free conservation as the theoretical property, with resetting as a practical add-on.

Referee Report

3 major / 6 minor

Summary. The paper introduces a finite-dimensional discretization of the incompressible Euler equations based on a Koopman representation of volume-preserving diffeomorphisms on half-densities. The discretized velocity constraint X ∈ im(Ā) is nonholonomic, and the authors treat it with the vakonomic variational principle, obtaining a reduced Lie–Poisson system in Lax form, Eq. (6.12): Ż = [Z^#, Z]. They show this system is isospectral, hence preserves Casimir invariants, and satisfies a discrete analogue of Kelvin's circulation theorem. A low-rank Clebsch momentum-map representation reduces the computational cost from O(F^2) to O(mF), and a Lie-trapezoidal time integrator preserves energy and the coadjoint orbit in the reset-free case. The practical algorithm includes a periodic resetting procedure to control dispersion. Numerical experiments include Taylor–Green vortex convergence, invariant checks, shielded and leapfrogging vortices, flows on surfaces, a 3D trefoil knot, and a Boussinesq Rayleigh–Taylor simulation.

Significance. The core geometric derivation is clean and largely self-contained: Theorem 6.1 derives the coadjoint equation from the vakonomic action, Lemma 6.1 gives the coadjoint action on so(F), and Theorem 8.1 shows the Clebsch ansatz satisfies the Lax equation exactly. The integrator analysis in Theorem 9.1 is rigorous and the numerical experiments are extensive. If the discrete system were shown to converge to the Euler equations in the continuum limit, this would be an important contribution to structure-preserving fluid simulation. Even without such a convergence theorem, the Lax formulation, the low-rank momentum-map representation, and the numerical evidence of reduced dispersion are valuable. The paper is also commendably explicit about its open questions, especially the continuum-limit issue in Section 12.

major comments (3)

[Section 12; Sections 5–6] The paper's central claim—that the vakonomic Lax system (6.12) is a discretization of the incompressible Euler equations (3.9)—is not established. Section 12 explicitly leaves open whether so(F) and im(Ā) converge to sdiff(M), and the numerical evidence in Fig. 12 is only for the steady Taylor–Green vortex. Without a consistency or convergence result for the general initial-value problem, Eq. (6.12) is a self-consistent isospectral flow on so(F) but not proven to approximate Eq. (3.9). I recommend either supplying such a result or reframing the contribution as a structure-preserving discrete fluid model whose relation to the Euler equations is empirical.
[Abstract; Section 10.3; Section 11.1.1] The abstract's 'machine-precision satisfaction of Casimir invariants' is true only in the reset-free limit. Algorithm 6 includes resets, all production experiments use resets (Table 2), and Section 11.1.1 reports a 1.6% energy drop over 5 s for TGV with resets; Fig. 16 shows Casimir jumps at each reset. Since resetting is a non-Hamiltonian heuristic (Sec. 10.3.2), the headline conservation claim should be qualified to the reset-free integrator, and the practical algorithm should be presented as approximately structure-preserving with quantified drift.
[Section 8; Section 12] The low-rank Clebsch ansatz (8.2) is exact on the invariant manifold rank(Z) ≤ 2m, but the paper does not quantify the approximation error for general initial data that are not low-rank, nor does it prove that the discrete analogue of the d−1 Clebsch-pair sufficiency of [83] holds in the finite-dimensional setting. The paper itself notes 'some variance in practice' (Sec. 12). This is not fatal, but it should be acknowledged explicitly in the main text rather than presenting the low-rank model as a closed reduced-order description with no approximation gap.

minor comments (6)

[Section 6] Heading typo: 'Eqations' should be 'Equations'.
[Section 5.4] The symbol V is used both for the finite element space and for its dimension. Use, e.g., dim V to avoid ambiguity.
[Figure 15 / Table 2] The reset-frequency parameter α in Fig. 15 is not defined in the table or caption; clarify whether it is a step count or a time interval.
[Section 10.3.1] The FTLE threshold S_f ≤ 1 is stated to 'consistently lead to good results' but no sensitivity analysis is provided; this is an empirical heuristic and should be labeled as such.
[Section 11.1.2] The comparison baseline 'Vorticity FEEC' is reconstructed via Reset for Casimir diagnosis. Please explain why this reconstruction is faithful to the LdA dynamics; otherwise the comparison in Fig. 14 is difficult to interpret.
[Section 7.1] The statement that LdA is 'not self-consistent' is normative; consider rewording to 'not variational in the same intrinsic sense' to avoid overstatement.

Circularity Check

1 steps flagged · score 3.0 of 10

Main Lax derivation is self-contained; the advertised discrete Kelvin theorem is a definitional restatement, and resetting plus the unproven continuum limit qualify the headline claims.

self definitional [Section 6.3.2, Theorem 6.2 (after Eq. (6.12))]
"A time-dependent discrete loop Y(t) is said to be transported by the flow of X(t) if it satisfies the Lax equation Ÿ(t)=[X(t),Y(t)]. ... Given a covector Z∈so(F)*, the circulation of Z along the discrete loop Y∈so(F) is just the evaluation pairing ⟨Z|Y⟩. So, a path Z satisfies Kelvin's circulation law ... if the circulation ⟨Z(t)|Y(t)⟩=⟨Z(0)|Y(0)⟩ is independent of t for all loops Y ... satisfying the Lax equation Ÿ(t)=[X(t),Y(t)]."

The theorem states that Z satisfies Ż=[X,Z] iff ⟨Z|Y⟩ is constant for every Y with Ÿ=[X,Y]. Because 'transported' is defined as satisfying the Lax equation and 'circulation' is defined as the pairing, the proof reduces to d/dt⟨Z|Y⟩=⟨Ż+ad*_X Z|Y⟩, which is zero exactly when the Lax equation holds (using ad*_X Z=-[X,Z]). The advertised discrete Kelvin theorem is therefore a restatement of Lax-pair invariance, not an independent consequence derived from the variational principle. It adds no content beyond the isospectral structure already present in Eq. (6.12).

full rationale

The core derivation in Sections 3–6 is self-contained: the vakonomic action (6.7), Lagrange multiplier, and sharp map produce the coadjoint equation (6.9) and the Lax form (6.12) without assuming the target result. The claimed Lie–Poisson structure and Casimir preservation are direct mathematical consequences of the Lax/isospectral structure and the coadjoint-orbit-preserving integrator (Theorem 9.1), not fitted inputs. Numerical verifications are self-consistency checks of exact invariants plus external benchmarks such as Taylor–Green and point vortices, so they are not circular predictions. Two acknowledged limitations, the open continuum-limit question in Section 12 and the resetting procedure in Section 10.3, are not circularity: the former leaves the Euler-discretization claim conditional, and the latter explicitly breaks exact Casimir conservation in the implemented algorithm. Self-citations [70] and [71] are used for implementation choices and heuristics rather than to justify the central derivation. The only definitional circularity is the 'discrete Kelvin theorem,' which is a restatement of Lax-pair invariance.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The core derivation uses only standard geometric mechanics and linear algebra; the main auxiliary inputs are the reset hyperparameters, the FTLE threshold, the choice of m Clebsch pairs, and the unproven confluence of the discrete sub-Riemannian structure with the continuous Euler equations. The 'invented entities' are mathematical constructs for stating conservation laws, not new physical degrees of freedom.

free parameters (4)

Reset period = Per-experiment: e.g., 2.29 s (TGV 128), 0.21 s (trefoil), 0.12 s (bunny); or FTLE-triggered
Chosen by hand per experiment (Table 2); controls how often the structure-preserving flow is interrupted. Frequent resets dissipate energy/Casimirs; infrequent resets accumulate dispersion.
FTLE reset threshold S_f = S_f ≤ 1
Empirical heuristic inherited from [71]; triggers reset of the Clebsch labels to limit stretching.
Number of Clebsch pairs m = m = d (2 or 3)
The paper notes theory requires only d−1 pairs [83], but uses m=d for practical robustness; this is an ad hoc modeling choice that restricts the rank of Z to 2m.
Time step Δt = varies per experiment (Table 2)
Standard discretization choice; implicit fixed-point solves are run to tolerance.

assumptions (5)

domain assumption The discrete Koopman representation imposes a nonholonomic constraint X ∈ im(Ā) that is the correct discrete analogue of the continuous constraint im(adv) ⊂ so(HD(M)).
Section 5/Example 5.1; if this constraint is not faithful, the resulting sub-Riemannian geodesics may not represent fluid motion.
domain assumption The finite-dimensional sub-Riemannian structure (SO(F), D, ⟨·,·⟩_K) yields geodesics that converge to continuous Euler solutions in the continuum limit.
Section 6.1 sets up the sub-Riemannian structure, but the continuum-limit convergence is left open in Section 12.
domain assumption FEEC spaces give pointwise divergence-free velocity and exact Hodge decomposition, making V_div = {u | div(u)=0}.
Section 10.2; if the discrete divergence operator does not faithfully represent the continuous one, the pressure projection is wrong.
ad hoc to paper The low-rank Clebsch ansatz Z = 1/2 Σ (λ_a μ_a^T − μ_a λ_a^T) captures the relevant dynamics of the full Lax equation.
Theorem 8.1 shows it is a closed subsystem, but not all solutions of Ẑ=[Z^#,Z] have rank ≤ 2m; the choice m=d is not proven sufficient.
standard math Standard results in variational calculus and sub-Riemannian geometry (e.g., [75]) justify the equivalence of vakonomic critical points and sub-Riemannian geodesics.
Used in Theorem 6.1 and Section 6.2.

invented entities (2)

Discrete loop Y ∈ so(F) transported by the flow Ẏ=[X,Y]
purpose: To state a discrete analogue of Kelvin's circulation theorem; circulation is defined as ⟨Z|Y⟩.
An algebraic construction internal to the paper; it is not a physical material loop and its connection to continuum circulation is asserted, not proven.
Accumulated gravity half-density g
purpose: Extends the vakonomic formulation to Boussinesq stratified flow by accumulating ∫g y along particle paths.
A numerical device introduced in Section 11.5; no independent experimental handle.

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Cite this review

Pith. "Pith review of Vakonomic Fluids." pith.science (2026). https://pith.science/paper/TIR6SILI

@misc{pith2026260718312,
  author       = {Pith},
  title        = {Pith review of: Vakonomic Fluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TIR6SILI}},
  note         = {Machine review of arXiv:2607.18312}
}

read the original abstract

We introduce a novel discretization of the incompressible Euler equations based on their interpretation as geodesic equations on the Lie group of volume-preserving diffeomorphisms. It is well known that encoding diffeomorphisms and their infinitesimal generators through a discretized Koopman representation places a nonholonomic constraint on discrete velocities, for which there is no consensus on a variational treatment. We show that taking the vakonomic perspective, as opposed to the usual perspective of Lagrange--d'Alembert, yields discrete fluid trajectories that remain geodesics on a (sub-)Riemannian manifold. In particular, the resulting vakonomic dynamics are Lie--Poisson and their solutions admit a discrete relabeling symmetry, leading to machine-precision satisfaction of Casimir invariants along with a discrete analogue of Kelvin's Circulation Theorem. Using an efficient momentum map representation based on low-rank Clebsch variables, we show that these vakonomic fluids behave stably and consistently even at low grid resolutions, leading to increased robustness and physical realism in the long term.

Figures

Figures reproduced from arXiv: 2607.18312 by the authors.

**Figure 2.** A cartoon of “variation space”: each point is a path [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Under the vakonomic variational principle (right), variation curves [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (37 more)

**Figure 4.** Figure 4: The geometry of the Lie group SDiff (𝑀). Here, the Lagrangian velocity 𝜑¤ at 𝜑 ∈ SDiff (𝑀) is right-translated to the identity id, yielding a representation by the divergence-free 𝑢® ∈ 𝔰𝔡𝔦𝔣𝔣(𝑀). 3.2 Fluids as Geodesics The variational formulation of the incompressible …

**Figure 5.** Figure 5: A depiction of the variational calculus on SDiff(𝑀). The flow map 𝜑 = 𝜑𝑡,𝜀 depends on time 𝑡 ∈ [0,𝑇 ] and a variational parameter 𝜖 ∈ [0, 1]. The SDiff(𝑀) elements id = 𝜑0,𝜖 and 𝜑𝑇 ,𝜀 denote fixed endpoints shared between all variations, while the vectors 𝜑¤𝑡,0, 𝜑˚𝑡,0…

**Figure 6.** Figure 6: An illustration of the Lie algebra 𝔰𝔬(HD (𝑀) ). Each vector field 𝑢® ∈ Γ(𝑇𝑀) 𝔡𝔦𝔣𝔣(𝑀) corresponds to a skew-adjoint Lie derivative operator adv𝑢® = − L𝑢® ∈ 𝔰𝔬(HD (𝑀) ), but not every skew-adjoint operator B ∈ 𝔰𝔬(HD (𝑀) ) takes this form. the space 𝔰𝔬(HD (𝑀)) of all sk…

**Figure 7.** Figure 7: Discretizing the advection operator adv : 𝔡𝔦𝔣𝔣(𝑀) → 𝔰𝔬(HD (𝑀) ). A divergence-free field 𝑢® ∈ V is identified with its coefficients u ∈ 𝑉div. The Galerkin projection A𝑢® of adv𝑢® on the half-density basis defines the fully discrete advection map u ↦→ Au ∈ 𝔰𝔬(𝐹 ) via Au…

**Figure 8.** Figure 8: The nonintegrable distribution D defined by im(A¯) ⊂ 𝔰𝔬(𝐹 ). Righttranslation of X ∈ im(A¯ ) yields elements XR ∈ DR of the partial tangent space DR ⊂ 𝑇R𝑆𝑂 (𝐹 ) forming the distributional fiber at R ∈ SO(𝐹 ). Notice that this sharp map also contains the familiar press…

**Figure 9.** Figure 9: The relationship between the discrete advection [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 9.** Figure 9: The relationship between the discrete advection A¯ and its adjoint A¯ ∗ as a commutative diagram. Any coefficient vector u ∈ 𝑉div maps to a unique skew-adjoint operator A¯ u ∈ 𝔰𝔬(𝐹 ). Similarly, any coset Y = Z + im(A¯) ◦ ⊂ 𝔰𝔬(𝐹 ) ∗ represented by Z ∈ 𝔰𝔬(𝐹 ) ∗ /im(A¯) …

**Figure 10.** Figure 10: The pipeline of our method, Vakonomic Fluids. The auxiliary sym [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 10.** Figure 10: The pipeline of our method, Vakonomic Fluids. The auxiliary symplectic space (𝑃 𝑚,J) is identified with the dual Lie algebra 𝔰𝔬(𝐹 ) ∗ , on which the equations of motion take an identical form. There, the dynamics is a Lie–Poisson system evolving along the intersect…

**Figure 11.** Figure 11: Evolution of the 𝑥-component of 𝝀 over time for the Taylor-Green Vortex initial condition. Above: Without resetting, 𝝀 develops dispersive artifacts emerging from the inability to resolve the shearing present in the flow map. Lower: With resetting every 120 frames (0.…

**Figure 14.** Figure 14: Conserved quantities for the 2D Laplacian-eigenfunction vortices, [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]

**Figure 12.** Figure 12: Convergence of the method with FEEC–IGA B-spline discretization spaces, with quadratic (𝑝 = 2, left) and cubic (𝑝 = 3, right) spatial accuracy. Observe that temporal discretization error dominates over longer time horizons, yielding second-order convergence in both …

**Figure 13.** Figure 13: Vorticity initialized as a random superposition of the first 20 eigen [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]

**Figure 16.** Figure 16: Casimirs tr(Z 𝑘 ) and energy for the first 400 frames of Taylor-Green Vortex simulation. Above: No resetting. Below: Resetting every 120 frames. The field u(𝑥, 𝑦, 𝑡) = uTGV(𝑥, 𝑦) solves the incompressible Euler equations for all 𝑡: the velocity, and hence the vorticit…

**Figure 13.** Figure 13: We attribute the difference to the structure each scheme [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]

**Figure 15.** Figure 15: Reset frequency, denoted as 𝛼, swept from every 50 to every 200 steps, probing how our solver behaves under different reset rates. Top row: the quadratic and quartic Casimir power traces. As resets become more frequent, the Casimir behavior converges toward the fini…

**Figure 17.** Figure 17: A random divergence-free velocity field u0 is selected to initialize matrix vectors { (𝝀ˆ , 𝝁ˆ ) }𝑚 𝑖=1 through the matrix Y(0) = Au0 . Then the variables { (𝝀ˆ , 𝝁ˆ ) }𝑚 𝑖=1 are advected alongside with the Clebsch variables used during the simulation { (𝝀, 𝝁) }𝑚 𝑖=1…

**Figure 19.** Figure 19: Evolution of the Shielded Taylor Vortices over time. [PITH_FULL_IMAGE:figures/full_fig_p026_19.png]

**Figure 20.** Figure 20: 2D vortex leapfrogging marathon experiment, comparing our vakonomic formulation against the functional-fluids-on-surfaces FEEC baseline. Both methods are run at 128 × 128 resolution with timestep Δ𝑡 = 1/48 s and quadratic spatial accuracy; for our method, the flow map…

**Figure 20.** Figure 20: 2D vortex leapfrogging marathon experiment, comparing our vako [PITH_FULL_IMAGE:figures/full_fig_p027_20.png]

**Figure 19.** Figure 19: Evolution of the Shielded Taylor Vortices over time. vortex is prescribed through its vorticity using the radial profile 𝜔(𝑟) = 1 𝑎 2 − 𝑟 2 𝑎 2 exp 1 2 1 − 𝑟 2 𝑎 2 , 𝑎 = 0.3, where 𝑟 is the radial distance to the vortex center. This profile has a positive …

**Figure 21.** Figure 21: Conserved quantities for the leapfrogging vortices, each plotted [PITH_FULL_IMAGE:figures/full_fig_p027_21.png]

**Figure 22.** Figure 22: Point Vortices on an Oblate Spheroid. The sphere is made 50% transparent to illustrate vortices on the reverse side. 11.2.3. Six Vortices on an Oblate Spheroid. Another interesting test monitors the behavior of multiple point vortices on a surface with positive secti…

**Figure 23.** Figure 23: Vorticity evolution over time for the double shear layer experiment. [PITH_FULL_IMAGE:figures/full_fig_p028_23.png]

**Figure 21.** Figure 21: Conserved quantities for the leapfrogging vortices, each plotted as relative error against the initial value at 𝑡 = 0. Top row: the quadratic and quartic Casimir power traces. Bottom row: energy and enstrophy. Here the Casimirs follow the same trend as the kinetic en…

**Figure 24.** Figure 24: Above: Vorticity evolution on two spheres of different resolution, initialized by a projected ABC flow. Below: The finite-time Lyapunov exponent of the flow on the higher resolution sphere visualized. where 𝑎 = 10 is a scale parameter. We initialize this projected AB…

**Figure 23.** Figure 23: Vorticity evolution over time for the double shear layer experiment. Our method (above) is compared to Azencot et al. [10] (below). 11.4. Flows on Surfaces. Finally, to illustrate our method’s ability to perform larger scale fluid simulation experiments, we run a numb…

**Figure 25.** Figure 25: A flow on a helicoid; initialized by a projected ABC flow. [PITH_FULL_IMAGE:figures/full_fig_p029_25.png]

**Figure 25.** Figure 25: A flow on a helicoid; initialized by a projected ABC flow. in [PITH_FULL_IMAGE:figures/full_fig_p030_25.png]

**Figure 26.** Figure 26: A flow on a Stanford bunny; initialized once again by a projected [PITH_FULL_IMAGE:figures/full_fig_p029_26.png]

**Figure 26.** Figure 26: A flow on a Stanford bunny; initialized once again by a projected ABC flow. 11.4.4. Flow on a Bunny. Finally, to conclude, we run a fluid simulation on a Stanford Bunny mesh with 85.7k vertices; this mesh has more intricate geometry than our other examples. We initial…

**Figure 27.** Figure 27: Rayleigh–Taylor instability of a stratified fluid under the Boussinesq [PITH_FULL_IMAGE:figures/full_fig_p029_27.png]

**Figure 27.** Figure 27: Rayleigh–Taylor instability of a stratified fluid under the Boussinesq approximation. A heavier fluid (white ink) sits atop a lighter fluid (blue) and falls into it, developing the characteristic plumes and intricate mixing patterns as the flow evolves. Resolution 25…

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Pith tools

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