# A fully semi-Lagrangian discretization for the 2D Navier--Stokes equations in the vorticity--streamfunction formulation

## TL;DR

This paper introduces a semi-Lagrangian numerical method for 2D incompressible Navier--Stokes equations that ensures unconditional stability and simplifies computations by avoiding complex linear system solutions.

## Contribution

It presents a novel semi-Lagrangian discretization for both advection and diffusion in the vorticity-streamfunction formulation, enhancing stability and computational efficiency.

## Key findings

- Method achieves unconditional stability.
- Numerical experiments validate accuracy on classical benchmarks.
- Simplifies boundary condition implementation.

## Abstract

A numerical method for the two-dimensional, incompressible Navier--Stokes equations in vorticity--streamfunction form is proposed, which employs semi-Lagrangian discretizations for both the advection and diffusion terms, thus achieving unconditional stability without the need to solve linear systems beyond that required by the Poisson solver for the reconstruction of the streamfunction. A description of the discretization of Dirichlet boundary conditions for the semi-Lagrangian approach to diffusion terms is also presented. Numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method.