Conservative Integrators for Vortex Blob Methods

TL;DR

This paper develops conservative second-order integrators for vortex blob models of Euler's equations, demonstrating their accuracy and superior conservation properties compared to standard methods through theoretical proofs and numerical experiments.

Contribution

It introduces a novel family of implicit conservative integrators using the Discrete Multiplier Method for vortex blob models, with proven properties and improved conservation performance.

Findings

Proved conservative properties and second-order convergence of the integrators.

Numerical experiments confirm the integrators' accuracy and conservation advantages.

Implicit integrators outperform explicit methods in preserving conserved quantities.

Abstract

Conservative symmetric second-order one-step integrators are derived using the Discrete Multiplier Method for a family of vortex-blob models approximating the incompressible Euler's equations on the plane. Conservative properties and second order convergence are proved. A rational function approximation was used to approximate the exponential integral that appears in the Hamiltonian. Numerical experiments are shown to verify the conservative property of these integrators, their second-order accuracy, and as well as the resulting spatial and temporal accuracy of the vortex blob method. Moreover, the derived implicit conservative integrators are shown to be better at preserving conserved quantities than standard higher-order explicit integrators on comparable computation times.