# A gPAV-Based Unconditionally Energy-Stable Scheme for Incompressible Flows with Outflow/Open Boundaries

## TL;DR

This paper introduces an unconditionally energy-stable numerical scheme for simulating incompressible flows with open boundaries, combining gPAV and velocity-correction strategies to ensure stability and efficiency even with large time steps.

## Contribution

The paper develops a novel gPAV-based scheme that guarantees unconditional energy stability for incompressible flows with open boundaries, simplifying boundary treatment and enabling large time step simulations.

## Key findings

- Proven energy stability independent of time step size.
- Effective handling of open boundary conditions with severe backflow.
- Stable and accurate simulations at large time steps.

## Abstract

We present an unconditionally energy-stable scheme for approximating the incompressible Navier-Stokes equations on domains with outflow/open boundaries. The scheme combines the generalized Positive Auxiliary Variable (gPAV) approach and a rotational velocity-correction type strategy, and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions. The discrete energy stability of the proposed scheme has been proven, irrespective of the time step sizes. Within each time step the scheme entails the computation of two velocity fields and two pressure fields, by solving an individual de-coupled Helmholtz (including Poisson) type equation with a constant pre-computable coefficient matrix for each of these field variables. The auxiliary variable, being a scalar number, is given by a well-defined explicit formula within a time step, which ensures the positivity of its computed values. Extensive numerical experiments with several flows involving outflow/open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes.