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Hamiltonian Two-Way Coupling of Nonlinear Waves and 3D Flows

This paper introduces a Hamiltonian-based, nonlinear, and dispersive 2D wave model that enables canonically consistent two-way coupling with 3D fluid solvers, effectively eliminating interface artifacts while significantly improving accuracy and computational efficiency compared to existing methods.

Original authors: Sinan Wang, Ruicheng Wang, Taiyuan Zhang, Fan Feng, Jinjin He, Yuchen Sun, Zhiqi Li, Bo Zhu

Published 2026-08-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sinan Wang, Ruicheng Wang, Taiyuan Zhang, Fan Feng, Jinjin He, Yuchen Sun, Zhiqi Li, Bo Zhu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to film a massive ocean scene for a movie. You need the water to look real, with waves crashing against a ship's hull, splashing over a submarine, or rippling as a seaplane lands. To make this look convincing on a computer, scientists use complex simulations that calculate how every drop of water moves. However, simulating an entire ocean with this level of detail is impossible; the computer would run out of memory and time long before the scene finished. The standard solution has been to simulate only a small box of water around the action in high detail, while using a much simpler, cheaper model for the vast ocean surrounding it. The problem is that these two models have historically spoken different languages. The detailed 3D model captures the chaotic, steep, and powerful nature of real waves, while the simple 2D model used for the open ocean often treats waves as gentle, predictable ripples that cannot bend or steepen. When these two mismatched simulations meet at the edge of the box, the waves often reflect back unnaturally or create visible seams, breaking the illusion of a single, continuous body of water.

A team of researchers at Georgia Institute of Technology and Dartmouth College has developed a new way to bridge this gap, allowing the detailed 3D simulation and the wide 2D ocean model to talk to each other without creating these visual errors. Their approach replaces the old, simplified ocean model with a new one that can handle the same complex, steep, and interacting waves as the detailed 3D solver. By ensuring both sides of the simulation understand the same physics of how waves grow, steepen, and travel at different speeds depending on their size, the researchers created a system where the transition between the small, detailed area and the vast, open ocean is seamless. This allows for the creation of large-scale water scenes, from battleships cutting through heavy seas to submarines surfacing, where the water behaves consistently everywhere, without the computer needing to calculate every single drop across the entire ocean.

The core of this achievement lies in fixing a fundamental mismatch in how computer graphics has traditionally handled water. For decades, the industry has relied on two main types of simplified wave models for the open ocean. One model treats water as a shallow, non-dispersive fluid, meaning all waves travel at the same speed regardless of their size, which works well for rivers but fails to capture the unique behavior of deep ocean swells. The other model treats waves as perfectly linear, meaning they pass through each other without changing shape or interacting, which is accurate for calm seas but falls apart when waves become steep or crash. When these simplified models are coupled with a high-fidelity 3D solver that naturally produces steep, interacting waves, the result is a jarring disconnect. The 3D side sends out complex, steep waves, but the 2D side cannot understand them, causing the waves to bounce back or distort at the boundary.

To solve this, the researchers introduced a new 2D wave model based on a mathematical framework known as the Zakharov formulation. In plain terms, this framework treats the water surface not just as a height map, but as a system with two linked properties: the height of the wave and the speed of the water moving along the surface. By tracking both of these properties together, the model can naturally reproduce the way real waves steepen, interact, and change speed based on their size. This is a significant departure from previous methods that had to choose between speed and accuracy. The new model captures the complex, nonlinear behavior of real ocean waves while still running fast enough to be used in large scenes. It achieves this by using a specific mathematical expansion that breaks down the complex wave interactions into a series of steps that a computer can solve efficiently using standard grid calculations, avoiding the need for complex, slow mesh tracking that previous high-accuracy methods required.

The researchers tested this new system by coupling it with a standard 3D fluid solver in a localized box. They ran a series of experiments to see how well the two sides communicated. In one test, they sent a wave packet from the 3D box out into the 2D ocean. In previous systems, this often resulted in the wave changing shape or reflecting back at the boundary. In their new system, the wave traveled smoothly across the interface, maintaining its shape and speed perfectly. They also tested the system with a boat moving through the water. A boat creates a specific V-shaped wake, known as a Kelvin wake, which has a distinct pattern of waves that spread out behind it. Older methods often failed to reproduce this pattern correctly or created strange ripples behind the boat where the 2D and 3D models met. The new method produced a clean, continuous wake that flowed seamlessly from the detailed 3D area into the open 2D ocean, matching the results of a full, high-resolution 3D simulation of the entire scene but running more than four times faster.

The team demonstrated the power of their approach with several large-scale scenarios. They simulated a seaplane landing on the ocean, where the plane skims the surface and leaves a long, trailing wake that stretches far beyond the immediate area of the plane. They also simulated a battleship navigating through heavy, rough seas, a scenario where the waves are so steep they are close to breaking. In these difficult conditions, the system remained stable and the water looked consistent across the entire domain. Another test involved a submarine rising to the surface, creating a complex disturbance as it broke the water. In all these cases, the transition between the detailed 3D region and the surrounding 2D ocean was virtually invisible, with no obvious seams or unnatural reflections. The researchers noted that while the system is highly accurate, it does require a careful balance; if the waves become too extreme, the computer must slightly reduce the level of nonlinearity in the calculation to prevent the simulation from becoming unstable, but even with this adjustment, the results remained far superior to previous methods.

The success of this method suggests a new path forward for computer graphics and fluid simulation. By providing a way to couple a detailed, localized 3D simulation with a vast, efficient 2D ocean model without losing physical accuracy, the researchers have removed a long-standing bottleneck in creating realistic water scenes. The system allows for the simulation of massive ocean environments where the water behaves correctly everywhere, from the splash of a single drop to the swell of a distant wave, without the computational cost of simulating the entire ocean in high detail. This opens the door for more realistic and immersive visual effects in films and games, as well as more accurate scientific visualizations of ocean dynamics, all while running on standard computer hardware in a fraction of the time previously required. The work represents a shift from choosing between speed and realism to achieving both, ensuring that the water in a digital world can finally behave as it does in the real one.

Technical Summary: Hamiltonian Two-Way Coupling of Nonlinear Waves and 3D Flows

1. Problem Statement

Simulating large-scale free-surface flows in computer graphics often requires a hybrid approach: confining an expensive, high-fidelity 3D Navier–Stokes (NS) solver to a localized region (e.g., around a ship hull or falling object) while modeling the surrounding open ocean with a computationally cheaper 2D surface model.

The primary bottleneck in this 2D–3D decomposition is a fundamental mismatch in wave dynamics at the coupling interface. Existing efficient 2D wave models used in graphics fall into two categories, both of which fail to couple seamlessly with strongly nonlinear 3D solvers:

Linear Models (e.g., Airy theory): These capture correct dispersion (wavelength-dependent speed) but discard nonlinear interactions (e.g., wave steepening). When coupled to a nonlinear 3D solver, the 2D side cannot represent the amplitude-dependent dispersion or crest steepening generated by the 3D side, leading to visible spurious reflections and artifacts at the interface.
Non-Dispersive Models (e.g., Shallow Water Equations - SWE): These capture nonlinear advection but collapse the dispersion relation, treating all waves as traveling at the same speed regardless of wavelength. This fails to reproduce the distinct dispersive behavior of real oceans.

Prior attempts to solve this, such as using Boundary Element Methods (BEM), recover nonlinear accuracy but suffer from prohibitive computational costs ( $O(N^3)$ ) and complex mesh tracking requirements, making them impractical for large-scale, real-time, or interactive applications.

2. Methodology

The authors propose a novel framework that couples a localized 3D NS solver with a nonlinear and dispersive 2D Eulerian wave model based on the canonical Zakharov formulation.

2.1 The 2D Wave Solver: Hamiltonian Spectral Approach

The core of the method is a 2D wave solver that evolves the canonical pair of surface elevation ( $\eta$ ) and surface velocity potential ( $\psi$ ) under the exact water-wave Hamiltonian.

Zakharov Formulation: The system is governed by Hamilton's equations where the Hamiltonian is the sum of kinetic and potential energy expressed entirely in terms of surface variables.
Dirichlet–Neumann Operator (DNO): To close the system, the method employs the Craig–Sulem expansion of the DNO. This allows the evaluation of the normal derivative of the velocity potential on the moving surface without solving the full 3D Laplace equation at every step.
High-Order Spectral (HOS) Expansion: The DNO is expanded into a Taylor series of homogeneous powers of $\eta$ . The authors retain terms up to the second or third order (HOS-2 or HOS-3), reducing nonlinear corrections to Fast Fourier Transforms (FFTs) and pointwise products on a regular grid. This achieves a computational complexity of $O(N^2 \log N)$ .
Controllable Nonlinearity ( $\varepsilon$ ): A scalar perturbation parameter $\varepsilon \in [0, 1]$ $ε \in [0, 1]$ modulates the nonlinear terms.
- $\varepsilon = 0$ : Reduces to linear Airy theory.
- $\varepsilon = 1$ : Recovers the fully nonlinear (truncated) Zakharov system.
- This allows users to trade stability for accuracy, lowering $\varepsilon$ in scenarios with steep, near-breaking waves to prevent numerical instability.
Time Integration: The solver uses an exact-linear Adams–Bashforth scheme. The linear part (dispersion) is integrated exactly in Fourier space to avoid stiffness, while the nonlinear forcing is advanced using a second-order Adams–Bashforth extrapolation.

2.2 Two-Way Coupling Mechanism

The coupling is designed to be canonically consistent, meaning information flows smoothly between the 2D canonical state $(\eta, \psi)$ and the 3D bulk state without breaking the physical symmetries.

2D $\to$ 3D (Lifting): The 2D state $(\eta, \psi)$ is lifted to the 3D embedding box boundary. Using the Craig–Sulem expansion, the method reconstructs the bulk velocity potential $\Phi$ at the flat reference plane ( $y=0$ ). This is then extended to depth layers using finite-depth harmonic kernels to provide depth-resolved velocity boundary conditions for the 3D solver's FAB (Fluxed Animated Boundary) layer.
3D $\to$ 2D (Folding): The 3D solver exposes its surface elevation $\eta_{3D}$ . Instead of directly overwriting the 2D state (which causes discontinuities), the method applies an implicit relaxation source to the kinematic equation of the Zakharov system. This source exponentially decays the 2D elevation $\eta$ toward $\eta_{3D}$ within a transition band. Crucially, the conjugate variable $\psi$ is reconstituted via an inverse DNO solve using the relaxed $\eta$ and the surface vertical velocity derived from the Hamiltonian evolution. This ensures the 2D state remains a valid canonical pair $(\eta, \psi)$ after relaxation, preventing spurious reflections.

3. Key Contributions

Nonlinear and Dispersive 2D Wave Model: A novel Eulerian wave model for graphics that simultaneously captures nonlinear wave steepening and correct dispersion. It evolves the canonical Zakharov pair at $O(N^2 \log N)$ cost, avoiding the $O(N^3)$ complexity and mesh operations of BEM.
Canonically Consistent Two-Way Coupling: A coupling strategy that leverages the Hamiltonian structure of the wave model. By treating the interface exchange as maps between canonical variables and using implicit relaxation, the method suppresses interface artifacts and ensures energy-consistent information transfer.
Continuously Tunable Nonlinearity: The introduction of the parameter $\varepsilon$ allows for a continuous family of solvers, enabling users to balance nonlinear fidelity against numerical stability on a per-scene basis.

4. Results and Validation

The method was validated against high-resolution 3D Eulerian references and prior coupling techniques across various benchmarks:

Accuracy: In wave propagation tests, the proposed model reduced mean wave-height error by 1.7–5 $\times$ compared to SWE, BEM, and Airy baselines. In Stokes wave tests, the fully nonlinear model ( $\varepsilon=1$ ) remained phase-locked to analytical solutions, whereas linear models drifted significantly.
Performance: The 2D solver runs >1000 $\times$ faster than BEM. In coupled scenarios (e.g., a single boat), the full system runs >4 $\times$ faster than a full-domain GPU NB-FLIP simulation of the same size.
Coupling Fidelity:
- Dispersion Matching: The method produced symmetric wave fields with minimal seam artifacts, whereas prior methods (SWE, Airy-based) exhibited asymmetry or spurious reflections.
- Complex Scenarios: The system successfully simulated a seaplane landing, a submarine surfacing, a battleship in heavy seas, and two chasing boats. In all cases, the 2D–3D interface remained largely free of visible artifacts, with wakes propagating smoothly across the boundary.
Stability: While the fully nonlinear model ( $\varepsilon=1$ ) is highly accurate, the authors note that for extreme forcing (e.g., near-breaking waves in the battleship scene), reducing $\varepsilon$ (e.g., to 0.2) is necessary to maintain stability.

5. Significance and Claims

The paper claims to address a long-standing structural mismatch in graphics fluid simulation. By introducing a 2D model that is both nonlinear and dispersive, the authors enable a 2D–3D decomposition pipeline that is practical, accurate, and largely free of interface artifacts.

The significance lies in the Hamiltonian structure of the coupling. Because the 2D solver evolves the canonical pair $(\eta, \psi)$ , the coupling directions (lifting to 3D and folding from 3D) become natural mathematical operations (Dirichlet–Neumann maps) rather than ad-hoc data transfers. This allows the system to handle strongly nonlinear interactions (like ship wakes and splashes) that previous linear or non-dispersive models could not support without visible degradation.

The authors position their method as a trade-off: they accept a modest increase in computational cost and a slight reduction in unconditional stability compared to linear solvers in exchange for the nonlinear accuracy and coupling fidelity required for realistic large-scale ocean simulations involving complex fluid-solid interactions. They acknowledge limitations, such as the inability to simulate overturning breaking waves (due to the single-valued height field) and the need for manual tuning of $\varepsilon$ in extreme scenarios, but assert that the method is practical for a broad range of production-style scenarios.