High order semi-Lagrangian methods for the incompressible Navier-Stokes equations

TL;DR

This paper introduces high-order semi-Lagrangian methods combining spectral element discretizations and exponential Runge-Kutta integrators for solving the incompressible Navier-Stokes equations, demonstrating effectiveness in convection-dominated scenarios.

Contribution

It develops and tests high-order semi-Lagrangian schemes for Navier-Stokes equations, extending existing methods with spectral and exponential integrator techniques.

Findings

Methods achieve high accuracy in space and time.

Effective for convection-dominated flows.

Numerical experiments confirm good performance.

Abstract

We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge-Kutta type. We discuss the extension of these methods to the Navier-Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.