Adaptive geometric numerical integration for point vortex dynamics

Area of Science:

• Computational physics
• Fluid dynamics
• Numerical analysis

Background:

• Hamiltonian dynamics of point vortices are crucial in fluid mechanics.
• Existing integration methods may struggle with long-term accuracy and efficiency.
• The Zhang and Qin scheme offers a basis for explicit symplectic integration.

Purpose of the Study:

• To develop and evaluate a variable stepsize integration method for Hamiltonian point vortex dynamics.
• To assess the method's ability to preserve conserved quantities and structural properties.
• To investigate the performance of the adaptive method in simulating the three-vortex problem, specifically exchange-scattering.

Main Methods:

• Implementation of a variable stepsize integration method based on the explicit symplectic Zhang and Qin scheme.
• Numerical simulation of the exchange-scattering phenomenon in a three-vortex system.
• Analysis of orbital symmetry and energy evolution during simulations.
• Long-term integration tests on various three-vortex configurations.

Main Results:

• The adaptive Zhang-Qin method is explicit and preserves the reversible structure of the flow.
• Numerical studies of the exchange-scattering phenomenon demonstrate the method's behavior.
• The method shows good efficiency and preservation of first integrals in long-term integrations.
• Symmetry of the orbit and energy evolution were analyzed for the exchange-scattering model.

Conclusions:

• The adaptive Zhang-Qin method is an efficient and reliable tool for simulating Hamiltonian dynamics of point vortices.
• The method demonstrates excellent preservation of first integrals, crucial for long-term stability.
• This adaptive approach offers advantages for studying complex vortex interactions, like exchange-scattering.