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REVIEW 4 major objections 5 minor 52 references

This paper claims that a charge simulation method can approximate multi-phase Mullins–Sekerka flow with triple junctions, mobile wall contacts, and multi-region phases, while conserving every bounded phase area to machine precision at the v

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2026-08-02 06:18 UTC pith:5RB2NTVF

load-bearing objection A genuine but incompletely validated extension: the multi-phase junction handling is new and the conservation proof is clean, but the only exact benchmark has no triple junction. the 4 major comments →

arxiv 2607.12759 v2 pith:5RB2NTVF submitted 2026-07-14 math.NA cs.NAmath.AP

A Structure-Preserving Method of Fundamental Solutions for the Multi-Phase Mullins-Sekerka Flow

Tokuhiro Eto This is my paper
classification math.NA cs.NAmath.AP MSC 65M8035R3753E1080A22
keywords Mullins-Sekerka flowcharge simulation methodmethod of fundamental solutionstriple junctionsstructure-preserving discretizationphase area conservationNeumann boundary conditioncurve networks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that a charge simulation method, a variant of the method of fundamental solutions, can discretize multi-phase Mullins–Sekerka flow in the plane and in a Neumann half-plane while preserving the flow's structure. The scheme represents each phase's chemical potential as a combination of fundamental solutions centered off the interface, so no bulk mesh or singular integrals appear, and it enforces the Gibbs–Thomson and continuity conditions by least-squares collocation, with triple junctions handled geometrically through the Herring–Young angle condition. The central structural claim is that every bounded phase area is conserved at the velocity level to machine precision, independent of the field-solve residual, by an orthogonal projection onto the null space of the discrete area constraints. This matters because the continuous flow's main conserved quantities—phase areas—are then not corrupted by numerical drift, so long-time coarsening simulations can be trusted. The scheme is validated against an exact three-concentric-circle solution, showing roughly first-order convergence of the interface error, though no exact benchmark exists for triple-junction networks.

Core claim

The discovery claimed is that the multi-phase Mullins–Sekerka flow, including curve networks with triple junctions, phases occupying multiple disconnected regions, and mobile orthogonal wall contacts, can be approximated by a boundary-only charge simulation method that is structure-preserving in area. The constructed chemical potential in each region is a finite sum of logarithmic point-charge fundamental solutions, with a near principal charge and a far auxiliary charge placed outside each region; the Laplace equation is satisfied exactly away from the interface and the far-field decay is built in. All interface conditions are imposed by least-squares collocation, and the Herring–Young law

What carries the argument

The central object is the charge-simulation ansatz: each regional chemical potential is a difference of two fundamental solutions centered outside the region—a near charge at distance 1/sqrt(M_r) and a far dummy charge at distance M_r^{3/2}—which enforces harmonicity, the far-field decay, and scale invariance without bulk meshes or singular integrals. The structure-preserving mechanism is the orthogonal projector P = I − C^T(CC^T)^{-1}C, which turns the per-phase area-preservation rows into hard constraints so that every reconstructed velocity has exactly zero net flux through each bounded phase, regardless of the least-squares residual. In the half-plane, each source is reflected across the

Load-bearing premise

The load-bearing premise is that the off-curve fundamental-solution ansatz with its two pre-scaled charge distances can accurately represent the harmonic chemical potentials all the way up to triple junctions and wall contacts; the paper has no exact benchmark or analysis for junction networks, so if corner-like singularities dominate there, the reconstructed junction velocities would be biased and the three-circle convergence would not transfer.

What would settle it

Take a three-circle configuration with high resolution and compare the edge-center normal velocities against the exact solution: if the error stagnates rather than decreasing as N grows, the ansatz fails. A more targeted test: embed a nearly straight interface ending at a triple junction in a symmetric configuration where the exact equilibrium is known, refine only the junction-adjacent edges, and measure the junction-edge velocity error against a high-accuracy boundary-integral reference; if that error does not shrink under refinement, the curvature correction is not recovering the true curva

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • No bulk mesh is needed: only the interface curves are discretized, and the elliptic solve costs about O(N^2) assembly plus a small projected solve that converges in a handful of CGLS iterations.
  • Because phase areas are conserved at the velocity level independently of the field residual, long-time coarsening simulations will not confuse numerical drift with physical mass transfer.
  • The same construction handles triple junctions, phases made of several disconnected regions, and mobile 90-degree wall contacts, with the Neumann boundary condition imposed exactly by image charges.
  • In the exact three-concentric-circle benchmark, the computed interface error decreases under refinement with an observed order approaching first order, and the discrete interfacial energy decreases monotonically as the flow predicts.
  • A region that shrinks below the mesh scale can be removed with its phase-area target re-baselined, enabling simulation of coarsening with simple topological events.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the method holds beyond the tested configurations, the same charge-simulation ansatz could be extended to anisotropic surface tensions or to three-dimensional flows by swapping the logarithmic fundamental solution for the appropriate Green's function, though the scale-invariance argument for the charge placement is specific to two dimensions.
  • The first-order convergence seen in the no-junction test likely masks a higher-order spatial error away from junctions; separating the junction curvature correction from the field-solve error could reveal whether the one-sided correction is the limiting factor.
  • The paper leaves open the stability analysis of the curvature-corrected junction treatment; a linearized analysis around a stationary symmetric triple junction would provide a concrete test of whether the correction introduces spurious long-wavelength modes.
  • The exact velocity-level conservation suggests that coupling the projection with a variational or symplectic time integrator could promote conservation from the velocity level to the fully discrete polygon level, removing the small polygonal drift reported in the paper.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes a charge-simulation / method-of-fundamental-solutions discretization of the multi-phase Mullins–Sekerka flow in the whole plane and in a Neumann half-plane. The chemical potential in each region is represented by differences of fundamental solutions centered at off-curve charge points, so the Laplace equation and the far-field decay are satisfied by construction; the Gibbs–Thomson law and continuity of the potential are imposed by two-sided least-squares collocation, and per-phase area identities are enforced as hard constraints through a null-space projection of the reconstructed velocity. Triple junctions are updated geometrically by a 120-degree (or prescribed Young-angle) correction, the Neumann wall is imposed exactly by image charges, and mobile wall contacts are evolved with chart-based updates that keep the 90-degree contact condition exact. The paper proves continuous length dissipation and area conservation, solvability and exact velocity-level area conservation for the projected discrete system, and reports numerical experiments: a three-concentric-circle exact solution, several whole-space junction networks, a region-removal test, and half-space junction/wall configurations. The only quantitative convergence benchmark is the three-circle solution, which has no triple junctions and shows first-order interface-error decay (EOC 0.5–1.14).

Significance. If the claims are accepted, the scheme would be a genuinely useful boundary-only method for multi-phase Mullins–Sekerka flow: no bulk mesh, exact image treatment of the Neumann wall, structural area conservation by projection, and support for phases occupying multiple disconnected regions. The paper is transparent about its limitations: it explicitly states that the triple-junction and half-space convergence/stability analysis is open, that no closed-form moving-junction benchmark is available, and that the reported conditioning (κ2 ≈ 1e17–1e19) is not backed by a stability analysis. These are strengths of presentation. However, the core novelty — reliable treatment of triple junctions and mobile wall contacts — is currently validated only by constraint residuals, energy curves, and circle-fit diagnostics, with no independent quantitative reference. The exact benchmark is also taken from the authors' own earlier work [24]. Thus the significance is conditional: the structural and algebraic parts are sound, but the numerical evidence for the junction/wall capabilities is incomplete.

major comments (4)
  1. [§9.1, §9.4, §10] The only quantitative convergence test is the three-circle solution, which has I_T = 0; Remark 9.1 states this explicitly. All junction and wall experiments in §9.2–§9.4 are assessed through diagnostics (E_A, E_angle, circle-fit residual, energy) with no independent reference. The exact solution is from the authors' own previous paper [24], so the benchmark is not independent. Since the paper's central claim is that the scheme handles curve networks with triple junctions, mobile wall contacts, and multiple regions, and since §10 states that convergence/stability analysis for those cases remains open, the central claim is not yet quantitatively supported. A comparison with a high-resolution PFEM/phase-field solution on a junction network, or a manufactured solution with controlled junction motion, would be needed.
  2. [Remark 5.2] The curvature correction on the four junction-adjacent edges is load-bearing: it directly modifies the Gibbs–Thomson right-hand side (6.10a) at exactly the edges where the reconstructed velocity is most sensitive. Remark 5.2 reports that the uncorrected junction-edge curvature is about a factor of two below the corrected one-sided estimate uniformly in resolution — a large, resolution-independent modification. No consistency proof or convergence test is supplied for the corrected quotient, and the remark does not isolate the effect of the correction on the junction dynamics. Without evidence that the corrected quotient converges to the true curvature as h → 0, the junction velocity bias remains unknown.
  3. [§6, Eqs. (6.1)–(6.3); §7] The MFS ansatz assumes that the harmonic chemical potential can be approximated uniformly up to the interface by charges placed at distances 1/sqrt(M_r) and M_r^{3/2}. Near a 120-degree triple junction the harmonic field generically has corner singularities, and the basis contains no mechanism (charge grading, singular enrichment, or adaptive placement) adapted to that geometry. No convergence analysis or analytic junction test is provided, and the conclusion acknowledges the problem remains open. This is not merely a theoretical gap: every junction/wall experiment depends on the field reconstruction on junction-adjacent edges, so the unverified MFS behavior near singular points is a load-bearing assumption of the central claim.
  4. [§8, Prop. 8.2; §9.1] Velocity-level area conservation is a structural consequence of the construction, not a prediction of the physical model. Because the area rows C in (6.10c) are assembled from the same velocity representation and the projector P in (6.11)–(6.12) enforces C q = 0 by construction, J_p(q) = 0 is an identity under (ND). Proposition 8.2 therefore verifies that the projection is implemented consistently, rather than that the discretization conserves phase areas on its own. The reported E_A in Table 2 is the subsequent polygonal drift of the advanced curves, which is not controlled by the projection. Please state this distinction more prominently and report the velocity-level flux residual separately from E_A so readers can distinguish constraint enforcement from physical conservation error.
minor comments (5)
  1. [§6, Eq. (6.13)] The notation v_{i,j} is used both for the edge-center normal velocity and, in the vertex update, for the advancing velocity. Please clarify the averaging formula that transfers edge data to vertices, since the two quantities are different.
  2. [§9.1, Fig. 7] The caption shows an O(N^{-1}) reference slope, but Table 2 reports EOC 1.14 at N=128. Please explain the discrepancy between the reference slope and the observed EOC.
  3. [§7.4, Eq. (7.14)] The adaptive step parameters c_CFL and Δt_max are empirical. A short sensitivity study with respect to these constants would help the reader understand the robustness of the half-space runs.
  4. [§7.3, Table 1] The validated class for wall-touching regions is stated as 'one contiguous wall interval' after the table. This restriction is important and should appear earlier in §7.3, before the constraint rows are introduced.
  5. [§9.3] The region-removal criterion δ_N = c_δ h_N^2 is resolution-dependent and the area removed vanishes at the claimed rate, but the choice c_δ = 0.02 is not justified. Please provide a short discussion of how the threshold compares with typical edge lengths and how the results change with c_δ.

Circularity Check

3 steps flagged

Central 'structure preservation' and junction/wall-angle results are imposed constraints rather than predicted outcomes; the only exact convergence test has no triple junction and comes from the authors' own [24], though its formula is restated.

specific steps
  1. self definitional [Abstract; Section 8, Proposition 8.2 and its proof; Eqs. (6.10c), (6.11), (6.12)]
    "The discretization is structure-preserving in the sense that every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. ... By Remark 8.2, every iterate satisfies q=P y, hence Cq=CP y=0 since CP=0 ... By the assembly identity Cq=(J1(q),...,JIP-1(q))^T this is Jp(q)=0."

    C is assembled as the area rows (6.10c), and Section 8 states that its p-th row 'coincides with J_p up to an overall sign.' P is defined as I - C^T(CC^T)^{-1}C, so CP=0 holds by construction. Thus Proposition 8.2 only restates the constraint; the 'conservation' is an enforced identity, not a consequence of solving the Gibbs--Thomson/continuity system, and the E_A diagnostics measure the residual of an imposed constraint rather than a predicted physical property. The paper is transparent about this, but the result is definitional.

  2. self definitional [Section 6, triple-junction geometric update paragraph; Section 9.2, unequal-surface-tensions paragraph]
    "The triple junctions are updated geometrically, not as rows of the linear system ... We compute a new common junction position p∗k such that the directions am−p∗k pairwise enclose the equilibrium angles (120◦ in the equal-tension case). ... Because these angles are imposed by the geometric junction update of Section 6 (its 120◦ target generalized to the per-junction Young angles), this experiment checks their preservation rather than their spontaneous emergence from the discrete evolution."

    The junction angles are set directly by a Gauss--Newton relocation of the endpoints to the prescribed equilibrium angles. The reported diagnostics E_∠ (9.2) therefore measure how accurately the constraint solver meets its own target, not whether the discrete evolution produces Herring--Young balance as an emergent consequence. The Section 9.2 quotation explicitly concedes that the experiment 'checks their preservation rather than their spontaneous emergence.' This makes the triple-junction validation circular in the same by-construction sense as the area projection.

  3. self definitional [Section 7.4, on-manifold wall-contact chart, Eq. (7.13); Table 4]
    "Each wall contact is equipped with a chart (X,h), Xcontact=(X,0)^T, Xadj=(X,h)^T, so the contact edge is vertical and the 90◦ angle is exact. ... max wall-contact angle deviation from 90◦ 0 (machine)"

    The 90-degree wall-contact angle is imposed by construction through the chart (7.13), and the '0 (machine)' entry in Table 4 is the residual of that imposed constraint. Reporting it as a quantitative validation of boundary-contact handling is therefore a self-definitional identity: the geometric update guarantees the angle exactly, so the diagnostic cannot independently confirm that the scheme correctly predicts the physical contact-angle law.

full rationale

The paper is unusually transparent: it explicitly says areas are conserved 'by a null-space projection' and that the unequal-tension angle experiment 'checks their preservation rather than their spontaneous emergence.' Hence the main circularity is not a hidden fitted parameter but a set of self-definitional identities presented as structural properties. Proposition 8.2 follows immediately from CP=0 with C assembled from the same J_p functionals, and the angle diagnostics measure the exactness of constraint solvers rather than emergent physical behavior. The MFS construction itself is not circular: the charge placement (6.3) is a design choice imported from the author's earlier [23], and the far-field and Neumann properties are elementary; no uniqueness theorem from the authors is invoked to forbid alternatives. The convergence test of Section 9.1 is an exact radial solution from the authors' own [24], but the paper restates the ODE (9.5)-(9.6) and computes the reference by quadrature, so it is not a black-box self-citation and provides some independent grounding. However, that benchmark has I_T=0, and the junction/wall experiments have no independent reference; they are assessed by diagnostics that are exact by construction. Overall this is partial circularity of the by-construction kind, not a fabricated derivation: score 5.

Axiom & Free-Parameter Ledger

11 free parameters · 8 axioms · 0 invented entities

No new physical entities are postulated. The dummy charge points, image charges, and endpoint charts are numerical devices from earlier literature (Sakakibara-Yazaki [48], [23]) repurposed here, not new entities with falsifiable handles.

free parameters (11)
  • β (charge-point far-distance exponent) = 3/2
    Introduced in (6.3) for the auxiliary charge distance M^β; chosen according to the author's prior work [23, §4], not derived.
  • α (half-space local source offset) = 1.5
    In (7.7), principal source offset α r_{i,j}; stated as an implementation parameter; not derived.
  • c (whole-space time-step constant) = 0.5 in convergence test, 0.05 in other runs
    Fixed rule Δt = c N^{-2}; admitted in Section 9 as an empirical choice.
  • c_CFL (half-space Courant constant) = 6.25e-3
    Adaptive step (7.14); hand-set.
  • Δt_max (half-space step cap) = 1e-3
    Cap in (7.14).
  • β0 (equidistribution strength) = 10
    Used in the soft redistribution rows (7.9), (7.11); hand-set.
  • λ_D (motion-law penalty weight) = 0.1
    In (7.8), penalizes flux-balance residual D; chosen to keep Gibbs-Thomson/continuity fit dominant.
  • λ_reg (Tikhonov regularizer) = 1e-8
    Regularizer in (7.9) to make the owned-vertex block positive-definite.
  • γ (compatibility weight) = 1
    Soft-row weight in Table 1.
  • δ (region-removal threshold) = 0.02 in Section 9.3; δ_N = c_δ h_N^2
    Removal criterion (9.8); the resolution-tied form is an ad hoc choice.
  • 10N factor in uniform-distribution method = 10N
    Controls equidistribution strength in the bidiagonal least-squares system of Section 6.
axioms (8)
  • domain assumption The exact radially symmetric three-phase solution of [24, §8.1] is a valid solution of (2.1)
    Used as ground truth in Section 9.1; authored by the same first author and not independently verified here.
  • domain assumption Non-degeneracy condition (ND): rank C = IP-1
    Required for the projector (6.11) and for Propositions 8.1 and 8.2; asserted to hold, not proved.
  • domain assumption Chemical potential w is bounded and ∇w = O(1/|x|^2), so [23, Lemma 3] applies
    Used in Proposition 4.1 to control boundary terms at infinity.
  • domain assumption Curve network satisfies Assumptions 3.1-3.4 (no self-intersection, triple junctions only, surjective region-to-phase map, adjacent regions in different phases)
    Defines the admissible geometry the method applies to.
  • ad hoc to paper The MFS ansatz (6.1) with charges at distances 1/sqrt(M) and M^{3/2} converges for multi-phase domains, including near triple junctions and wall contacts
    No convergence analysis is given; relies on [23] for the two-phase case and on the numerical conditioning report in Remark 9.1.
  • domain assumption Smooth classical solution exists and has finite Dirichlet energy
    Propositions 4.1 and 4.2 assume (w, Γ) is smooth; well-posedness with triple junctions is open per Remark 2.5.
  • standard math Fundamental-solution superposition and image charges satisfy Laplace and Neumann conditions
    Basis for CSM via equations (6.1) and (7.4)-(7.5).
  • standard math Morrey embedding W^{1,p}(Ω) -> C^α(Ω) (Remark 2.3)
    Used to justify continuity of W^{1,p} solutions across interfaces.

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read the original abstract

A charge simulation method is applied to approximate the multi-phase Mullins--Sekerka flow in $\mathbb R^{2}$ and in a half-plane bounded by a Neumann wall. In the underlying mathematical model, interfaces driven by their curvature are coupled through a harmonic chemical-potential field. We use a charge simulation method, a variant of the method of fundamental solutions: each chemical potential is represented by fundamental solutions centered at charge points off the curve, so no bulk mesh is required. It treats curve networks separating several phases at triple junctions, including phases that occupy more than one region; on the half-plane boundary, the no-flux condition is imposed exactly by image charges, and mobile contacts stay orthogonal to the wall. The discretization is structure-preserving in the sense that every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. The proposed scheme is assessed through a convergence test against an exact three-concentric-circle solution.

Figures

Figures reproduced from arXiv: 2607.12759 by Tokuhiro Eto.

Figure 1
Figure 1. Figure 1: A curve network Γ in the case IP = 3, IR = 4, IS = 4, and IT = 2. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Vertex and edge indexing for closed and open polygonal curves Γh i . We also define a discrete variant of the normal vector field on Γh i by ⃗νh i,j :=  X⃗ i,j − X⃗ i,j−1 ⊥ ri,j for i ∈ N≤IS and 1 ≤ j ≤ Ni , 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Charge-simulation set-up near a polygonal curve Γh i . Using these notations, we now explain how to determine the coefficients c (r) p and Q (r) p,i,j . We note that the chemical potentials w (r) p sum up to zero for each r ∈ N≤IR , so we only need to determine the coefficients for 1 ≤ p ≤ IP −1. Since the additive constant c (r) p is region-wise, each region contributes one constant per phase. Thus, the n… view at source ↗
Figure 4
Figure 4. Figure 4: Post-hoc geometric triple-junction update: (a) after the free curve update; (b) Reset of the endpoints to the new common point ⃗p ∗ k . 7 Extension to Neumann boundary problem So far, we have explained the implementation of the fully discrete scheme in the whole space R 2 . In this section, we extend the proposed scheme to the half space H with boundary contacts. 7.1 Half-space Neumann problem First, we in… view at source ↗
Figure 5
Figure 5. Figure 5: illustrates this construction near the wall: the charge points (7.7) of each region are superposed with their mirror images ⃗y(r)∗ i,j , ⃗z(r)∗ i,j reflected across W (7.3), so that the homogeneous Neumann condition holds on W exactly (7.5). W ⃗νW Γ h i ⃗νh i,j ⃗y(r − i ) i,j ⃗z(r − i ) i,j ⃗y(r + i ) i,j ⃗z(r + i ) i,j ⃗y(r − i )∗ i,j ⃗z(r − i )∗ i,j ⃗y(r + i )∗ i,j ⃗z(r + i )∗ i,j X⃗ i,j−1 X⃗ i,j X⃗ ∗ i,… view at source ↗
Figure 6
Figure 6. Figure 6: On-manifold endpoint charts: (a) the triple-junction chart (7.12), with fixed ray offsets φk; (b) the wall-contact chart (7.13), with a vertical contact edge. Adaptive time step. Unlike the whole-space runs of Section 6, which use the fixed rule ∆t = c N −2 , we choose the time step adaptively in the half-space runs as ∆t = min  cCFL mini,j ℓi,j maxΓ |V | , ∆tmax , (7.14) where mini,j ℓi,j is the smalles… view at source ↗
Figure 7
Figure 7. Figure 7: Convergence for the three-circle test. (a) Computed mean radii R¯h i (t) (markers) against the exact Ri(t) (solid), N = 128. (b) Interface error eΓ(T) against N, log–log, with an O(N −1 ) reference slope. (c) Interfacial energy L(t)/L(0), N = 128. tenfold smaller c = 0.05 it is about ten times smaller. The interfacial energy L(t) decreases monotonically, consistent with the curve-shortening property (Propo… view at source ↗
Figure 8
Figure 8. Figure 8: Whole-space experiments (N = 32, c = 0.05, run to the near-stationary criterion (9.4)): snapshots at t = 0, the midpoint of the observed energy decrease, and T [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Area-error and energy diagnostics (9.1), (9.3) for the three-phase baseline in the first row of [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Intra-phase ripening of two dispersed phases in an unbounded matrix: disks of radii 0.90, 0.45, 0.30 (phase 1, orange) and 0.80, 0.50, 0.35 (phase 2, blue), at N = 48 per curve, ∆t = cN −2 with c = 0.05, removal threshold δ = 0.02. Top: snapshots at the initial time, after the second removal, and at the final near-stationary state. Bottom: the six region areas Ar(t) with the threshold δ; the normalized ph… view at source ↗
Figure 11
Figure 11. Figure 11: Half-space three-phase M-branch from a strongly asymmetric, unbalanced state (A2 : A1 = 7 : 10); N = 28, fixed horizon T ≈ 0.11. Top: snapshots at t = 0, T /2, T; bottom: area-error and energy diagnostics [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Half-space four-phase, three-junction network with three mobile wall contacts, relaxing from a strong perturbation; N = 28, integrated to T ≈ 0.30. Top: snapshots at t = 0, T /2, T; bottom: area-error and energy diagnostics. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗

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