A preprint describes a water-simulation method that combines a nonlinear, dispersive 2D wave model with a Hamiltonian-based two-way link to a localized 3D Navier-Stokes solver. In its numerical tests, the reported time-averaged mean wave-height error was roughly four times lower in shallow water and three to five times lower in deep water than with SWE and BEM. Compared with Airy DK, the reported reduction was about 2.4 times in shallow water and 1.7 times in deep water. The wave solver was also reported as about 1,000 times faster than BEM.
How the handoff works
At the center of the system is a Zakharov state for the waves, advanced with a Craig-Sulem DNO expansion. The same machinery maps the surface state to depth-resolved 3D boundary velocities and, after relaxation, reconstitutes a canonical wave state. In plain language, the wave calculation passes information into the detailed 3D region and receives information back, which is what makes the coupling two-way.
The accuracy gap
The evaluation included large-scale numerical scenarios involving a 2D to 3D interface, including battleship, submarine, boat and seaplane cases. It also included long-time mild Stokes-wave tests. For coupling comparisons, the methods were driven by the same 3D NB-FLIP solver under identical initial conditions.
The accuracy measure was the time-averaged mean difference between simulated and reference wave heights. In shallow water, the proposed model's reported mean error was roughly four times lower than the SWE and BEM results. In deep water, it was three to five times lower than those methods. Against Airy DK, the corresponding reductions were about 2.4 times in shallow water and 1.7 times in deep water. No confidence intervals or inferential uncertainty estimates were reported.
Phase behavior was another dividing line. In long-time mild Stokes-wave tests, the linear solver progressively drifted out of phase with the analytical curve, while fully nonlinear HOS-3 remained phase-locked to it.
A cleaner boundary and faster scenes
The interface test focused on the point where the 2D and 3D calculations meet. The proposed method produced a wave field that was nearly indistinguishable across the two regions, with smoother waveforms in the 3D region. The interface finding was primarily qualitative, with no aggregate artifact metric reported.
The reported speed results were similarly specific to the tests. The wave solver was about 1,000 times faster than BEM. In a single-boat scene, the proposed coupled method ran at 85.0 milliseconds per substep, compared with 363 milliseconds for pure GPU NB-FLIP. No repeated-run variability or uncertainty estimate was reported for the runtime comparison.
Tests at stronger nonlinearity
The model's treatment of nonlinearity was tested by varying epsilon, the paper's nonlinear-amplitude parameter. Error decreased monotonically as epsilon increased. Order 2 and Order 3 were nearly indistinguishable up to about epsilon 0.6, then separated, with Order 3 lower at epsilon 1.
The large-scale evaluation included battleship, submarine, boat and seaplane cases. Across simulated ship speeds, the reported wake angles matched cited real-world data and captured the Kelvin-Mach transition.
Where the method still falls short
The paper reports that the fully nonlinear setting at epsilon 1 was not unconditionally stable under strong 3D forcing near the breaking threshold, and demanding simulations used a reduced epsilon. The 2D height-field model cannot represent breaking or overturning waves. Coupling was not fully artifact-free in the most demanding scenes, where render-time masking was used to mitigate residual artifacts.
An ablation test examined two numerical choices. Disabling the low-pass state filter was followed by quick instability. Replacing IF-AB2 with first-order integrating-factor Euler was accompanied by an increase in mean and maximum error from 0.011501 and 0.051843 to 0.012752 and 0.053980, respectively.
Status of the work
The item is labeled a preprint in the supplied metadata. Its front matter identifies arXiv version 1 dated 25 August 2026 and also gives an ACM Transactions on Graphics citation for volume 45, issue 6, Article 179, with a December 2026 publication date. The Georgia Tech authors report support from NSF grants and an NVIDIA Academic Grant, and credit a Houdini education license for video animations.
Paper data and sources
Original title: Hamiltonian Two-Way Coupling of Nonlinear Waves and 3D Flows
Authors: Sinan Wang, Ruicheng Wang, Taiyuan Zhang et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: 10.1145/3842540
Original paper · Full text