An analytically divergence-free collocation method for the incompressible Navier-Stokes equations on the rotating sphere

TL;DR

This paper introduces a high-order, divergence-free collocation method based on radial basis functions for solving the incompressible Navier-Stokes equations on a rotating sphere, enabling efficient and accurate simulations.

Contribution

It presents a novel RBF-based collocation approach that ensures divergence-free velocity approximations and simplifies pressure recovery without additional Poisson solves.

Findings

Achieves high-order accuracy for velocity and pressure

Provides precise error estimates for semi-discretized solutions

Demonstrates efficiency through a numerical test case

Abstract

In this work, we develop a high-order collocation method using radial basis function (RBF) for the incompressible Navier-Stokes equation (NSE) on the rotating sphere. The method is based on solving the projection of the NSE on the space of divergence-free functions. For that, we use matrix valued kernel functions which allow an analytically divergence-free approximation of the velocity field. Using kernel functions which lead to rotation-free approximations, the pressure can be recovered by a simple kernel exchange in one of the occurring approximations, without solving an additional Poisson problem. We establish precise error estimates for the velocity and the pressure functions for the semi-discretised solution. In the end, we give a short estimate of the numerical cost and apply the new method to an experimental test case.