Numerical Simulations of Surface-Quasi Geostrophic Flows on Periodic Domains

TL;DR

This paper introduces a new numerical algorithm for simulating surface-quasi geostrophic flows, combining explicit Runge-Kutta time discretization with flux-corrected transport for space, achieving stability, accuracy, and flexibility on various meshes.

Contribution

A novel algorithm for SQG flow simulation that handles complex geometries without mesh restrictions and demonstrates stability, accuracy, and realistic turbulence modeling.

Findings

Scheme satisfies a discrete maximum principle under CFL condition

Achieves second-order spatial accuracy in practice

Successfully simulates turbulence with Kolmogorov energy decay

Abstract

We propose a novel algorithm for the approximation of surface-quasi geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed and the algorithm does not have restrictions on the mesh structure nor on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second-order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.