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Numerical Study of Reported Multiple Solutions to a Cauchy Problem for the 2d Incompressible Euler Equations
Allen Tesdall, Richard Sanders
TL;DR
Earlier work proposed that two regularizations of a singular incompressible Euler initial datum could yield distinct spiral-containing limits. This study tests that claim with high-resolution computations and symmetry checks using a large domain and nested grids. Most parameter pairs yield a one-spiral limit; one exceptional pair may allow two, but the authors cannot confirm it.
Problem
Earlier work suggests that two regularizations of singular Euler initial data may converge to distinct self-similar limits, one with one spiral and one with two.
Source: §1 · p. 3Method
Results
In almost every parameter pair examined, both problems appear to approach one-spiral limits; for Problem 2, only α = 0.95, θw = π/16 may approach a two-spiral limit, which remains unconfirmed.
Source: §6 · pp. 35–36Takeaways & Limitations
The possible two-spiral ϵ-limit for Problem 2 appears confined to a subtle connection between the problem parameters.
Source: §6 · p. 36Computations use a very large but finite domain and have discretization error, so numerical trajectories need not satisfy the exact scaling symmetry.
Source: §4.2.1 · p. 23
Abstract
from arXiv · showhide
Bressan and Shen (2021) jointly with Bressan and Murray (2020) proposed a simple example of a Cauchy problem on $\mathbb{R}^2$, characterized by two key parameters, intended to yield nonunique weak solutions of the incompressible Euler equations. They suggest that a limiting process applied to two specific approximations of their example initial data yields two distinct solutions. Here we provide evidence coming from a carefully designed numerical procedure to support if, or when, this intriguing claim is valid. Some of the numerical challenges encountered in the present study are: (i) the proposed initial data is unbounded in a neighborhood of the origin, (ii) the proposed problem is posed on all of $\mathbb{R}^2$, (iii) several essential symmetries present in the true solution must be captured, (iv) the proposed example depends on two parameters which may be critically linked. We demonstrate that an extremely large computational domain is mathematically necessary to capture the core symmetries of the exact free-space problem. Accordingly we implement a nested-grid strategy specifically designed for its particular suitability and efficacy for the problem at hand. Our strategy also allows us to achieve extreme grid refinement in the region where solution accuracy is most critical. Within our nested-grid framework, we design an underlying discretization scheme that is high-order, robust, and fast. We consider a broad collection of allowable parameter pairs, but among all of them we find only one that may satisfy the suggested claim.
1. Introduction
The paper tests numerical claims that two regularizations of singular Euler initial data may converge to distinct self-similar solutions. It develops large-domain, locally refined computations to assess those claims while preserving the free-space problem’s symmetries.
1. Introduction
- The initial vorticity is origin-symmetric, supported on two infinite wedges, and grows without bound near the origin along rays where ϕ is nonzero.Its radial dependence is r^-α, with α in (0, 2); the two regularizations differ in whether a small disk between the wedges has vorticity or zero vorticity.Source: §1 · pp. 2–3
- Bressan and Shen and Bressan and Murray suggest that two regularizations yield self-similar limits with one versus two spiral centers.They describe this possible difference as evidence of “incurable” ill-posedness.Source: §1 · p. 3
- Their example at α = 0.95 and ϵ = 0.004 displays one spiral for the first regularization and two for the second.The displayed computations use a single uniform grid on [−0.2, 0.2] × [−0.2, 0.2].Source: §1 · pp. 3–4, Figure 1 · p. 4
- The study uses high-resolution numerical solutions of Problems 1 and 2 to assess the proposed nonuniqueness of their ϵ-limits.The problems are posed on R^2, but numerical solutions use bounded domains; the authors therefore use a very large domain to reduce far-field boundary effects and nested grids for local refinement.Source: §1 · p. 4
- The authors use a high-order scheme with nested-grid refinement near the origin, alongside a very large spatial domain to limit boundary effects.They also examine scaling symmetries and the self-similarity of ϵ-limits before evaluating the computed solutions.Source: §1 · pp. 4–5
2. Scaling basics and similarity
The section derives scaling relations for the regularized Euler problem and explains how they connect to self-similar limits. It uses those relations to identify observable symmetry checks, including spiral-center trajectories, for testing numerical solutions.
2. Scaling basics and similarity
- The angular factor ϕ is a nonnegative, π-periodic function, while α ∈ (0, 2) controls the initial vorticity’s radial dependence.The initial data is singular at the origin along rays where ϕ is nonzero.Source: §2 · p. 5
- Scaling relates solutions at different ϵ values: a solution for one fixed ϵ₀ generates solutions for arbitrary positive ϵ.The rescaling normalizes the initial-value problem to ϵ = 1.Source: §2 · p. 6
- The proposed self-similar limits are distinct: one has a single spiral and the other exactly two, according to Bressan and Shen and Bressan and Murray.A spiral is defined here as an isolated point of maximum or infinite vorticity.Source: §2 · p. 7
- Whether the suggested self-similar limits can be confirmed numerically is the study’s principal question.The section connects the scaling law to self-similar variables and the corresponding free-space boundary-value problem.Source: §2 · pp. 7–8
- For Problem 2, the authors test the proposed two-spiral limit by tracking the peak-vorticity location over time and checking whether its scaled path matches the expected symmetry.Antipodal symmetry requires two distinct spiral locations to be symmetric about, and not at, the origin.Source: §2 · p. 8
3. The numerical methods
The numerical method discretizes the vorticity conservation law with a second-order finite-volume scheme and advances it using Heun’s method. Implementation combines multigrid inversion with nested grids to resolve the large spatial domain and test boundary and discretization effects.
3.2. Implementation specifics.
- A multigrid V-cycle inverts the discrete Laplacian; the single-grid tests use L = 0.2 and stop when the residual max-norm falls below 5·10^-7.The reported implementation uses at most six cycles, independent of grid refinement.Source: §3.1 · p. 9, §3.2 · p. 12
3.1. Discretization generalities.
- The vorticity conservation law is discretized with finite-volume upwinding using edge velocities that satisfy a discrete divergence-free condition.The stream-function formulation yields the discrete analogue of ∇·u = 0.Source: §3.1 · p. 9
- Minmod reconstruction gives a generically second-order spatial method, advanced in time with Heun’s two-step method.Each Heun step requires two numerical Laplacian inversions.Source: §3.1 · pp. 10–11
- The second-order scheme satisfies a maximum principle when the time step meets a CFL-like restriction.The proof rewrites the predictor and corrector as convex combinations.Source: §3.1 · pp. 11–12
3.2. Implementation specifics.
- Most results use nested grids with L0 = 64 and generally refine to L0/2^9 = 0.125, testing far-field boundary effects and discretization accuracy.Each finer grid halves the spatial extent, with boundary data interpolated from the preceding grid.Source: §3.2 · pp. 12–13
4. Numerical results
The numerical results examine how the twin spirals evolve as ϵ decreases across parameter choices, including extended trajectories at selected values of α and θw. They show collapse toward a single spiral in the baseline and α=1.4 cases, while the extended trajectory at α=0.95 and θw=π/16 indicates a possible attracting self-similar solution with a nonzero spiral center.
4.1. Defining the initial data.
- The initial data in leave ϕ unspecified; the study uses a π-periodic tent function symmetric about the x-axis, with data depending on α, ϵ, and θw.The resulting initial data are bounded and discontinuous, with a discontinuity at r = ϵ.Source: §4.1 · p. 13
4.2. The numerical solutions.
- Computations use nested grids over [−64, 64]^2, with 4096^2 cells per subgrid and ϵ/∆x ≥ 4.096 to resolve the shrinking core.The ratio provides 8.192 grid cells across the disk of radius ϵ.Source: §4.2 · p. 15, §4.2.1 · p. 18
4.2.1. Baseline parameter calculations
- At fixed t, Problem 2’s twin spirals merge as ϵ decreases; at ϵ = 0.000125, both problems’ solutions appear as similar single spirals.The calculations indicate convergence to a one-spiral solution for fixed t > 0.Source: §4.2.1 · pp. 18, 21, Figure 5 · p. 19, Figure 6 · p. 20
4.2.2. The effect of initial data parameter α.
- Larger α makes the twin spirals orbit the origin more rapidly and closely; for α = 1.05 they essentially merge by ϵ = 0.0005, while α = 1.4 shows similar behavior earlier.Rotation and collapse are so rapid for α = 1.05 and 1.4 that trajectories for ϵ = 0.00025 and 0.000125 are omitted.Source: §4.2.2 · pp. 26–27
4.2.3. The effect of initial data parameter θw.
- Larger θw makes the twin spirals orbit the origin more rapidly and closely; for θw = π/4 and ϵ = 0.004, one spiral circles about eight times in three time units.Its distance from the origin eventually decreases, and the results indicate collapse to a single spiral as ϵ tends to zero for fixed t > 0.Source: §4.2.3 · p. 29
- At θw=3π/16, twin spirals are practically merged by ϵ=0.0005; at θw=π/4, the two-to-one transition occurs between ϵ=0.004 and 0.002.These results are part of a set in which both Problem 1 and Problem 2 numerically converge to a single spiral.Source: §4.2.3 · p. 30
- At θw=π/16 and α=0.95, extending the smallest-ϵ trajectory to t=20 gives the first indication of a possible attracting self-similar solution with y* ≠ 0.The computed xϵ(t)/t^(1/α) values appear to approach horizontal asymptotes; trajectories over t ≤ 10 had not ruled out a twin-spiral limit.Source: §4.2.3 · pp. 30–31, Figure 17 · p. 31
- At θw=π/16 and α=1.4, the twin spirals collapse to a single spiral as ϵ tends to zero for any fixed t > 0.The conclusion is supported by the computed time-trajectories and the visualized scaling relation.Source: §4.2.3 · p. 31, Figure 18 · p. 32
5. Discussion
Across nearly all tested parameter pairs, the computations indicate that Problem 2 tends toward a single spiral; the case α = 0.95, θw = π/16 remains inconclusive. Small uniform-grid computations fail the scaling-symmetry test, while the authors report that a much larger domain suffices.
5. Discussion
- For all but α = 0.95, θw = π/16, Problem 2 appears to converge to a single-spiral solution; the exceptional case neither confirms nor refutes a two-spiral limit.The trajectories in that case are close to straight and move at approximately the expected speed.Source: §5 · p. 32
- Bressan and Shen and Bressan and Murray report two distinct limits, but their cited contour plots do not establish that the depicted solutions are limits or self-similar.Their reported plots do not specify parameter values.Source: §5 · pp. 32–33
- The computations using the Matlab code show decreasing separation between twin spiral centers as ϵ decreases, but do not establish a two-spiral ϵ-limit.The code uses a first-order finite-difference scheme on a single uniform grid; the tested domain was enlarged to [−0.2, 0.2] × [−0.2, 0.2].Source: §5 · pp. 33–34
- The exact free-space trajectories obey a scaling identity, which the small-domain calculations fail to reproduce.The comparison rescales trajectories across ϵ values; the expected curves would overlay if the computed solutions respected the symmetry.Source: §5 · pp. 34–35
- Neither small-domain computation obeys the predicted scaling overlay; the authors report that L0 = 64 provides a sufficiently large domain to satisfy the symmetry adequately.The small-domain comparison uses a uniform grid on [−0.2, 0.2] × [−0.2, 0.2], whereas their nested-grid solutions use [−64, 64] × [−64, 64].Source: Figure 20 · p. 34, §5 · p. 35
6. Conclusion
The two regularizations converge strongly to the same limit as ϵ tends to zero. Problem 2 has one potentially exceptional parameter pair, but the authors cannot confirm a two-spiral limit there.
6. Conclusion
- The two regularizations converge strongly to the same limit as ϵ tends to zero.The authors report no evidence of self-similar two-spiral solutions in almost every parameter set considered.Source: §6 · p. 35
- The computations use a high-resolution finite-volume scheme with extreme grid refinement on a very large domain to satisfy the required scale symmetry accurately.The authors identify self-similarity and the free-space setting as central to the earlier analysis.Source: §6 · p. 35
- Problem 1 appears to approach a single-spiral solution for every tested case, and Problem 2 does so except possibly at α = 0.95, θw = π/16.The authors cannot confirm or refute a two-spiral self-similar limit in that exceptional case; if present, it would indicate a subtle parameter connection.Source: §6 · p. 36
CRediT authorship contribution statement
Allen Tesdall and Richard Sanders are listed as equal contributors.
CRediT authorship contribution statement
- Allen Tesdall and Richard Sanders are credited with equal contribution.Source: CRediT authorship contribution statement · p. 36