GePUP-ES: High-order Energy-stable Projection Methods for the Incompressible Navier-Stokes Equations with No-slip Conditions

TL;DR

This paper introduces high-order, energy-stable projection methods for the incompressible Navier-Stokes equations that ensure divergence decay and energy decrease, suitable for finite volume/difference schemes with high accuracy.

Contribution

It proposes GePUP-E and GePUP-ES formulations that are energy-stable, divergence-decaying, and equivalent to no-slip INSE, with high-order algorithms for improved numerical solutions.

Findings

Methods achieve exponential divergence decay.

Algorithms demonstrate fourth-order accuracy in velocity.

Numerical tests confirm theoretical properties.

Abstract

Inspired by the unconstrained PPE (UPPE) formulation [Liu, Liu, & Pego 2007 Comm. Pure Appl. Math., 60 pp. 1443], we previously proposed the GePUP formulation [Zhang 2016 J. Sci. Comput., 67 pp. 1134] for numerically solving the incompressible Navier-Stokes equations (INSE) on no-slip domains. In this paper, we propose GePUP-E and GePUP-ES, variants of GePUP that feature (a) electric boundary conditions with no explicit enforcement of the no-penetration condition, (b) equivalence to the no-slip INSE, (c) exponential decay of the divergence of an initially non-solenoidal velocity, and (d) monotonic decrease of the kinetic energy. Different from UPPE, the GePUP-E and GePUP-ES formulations are of strong forms and are designed for finite volume/difference methods under the framework of method of lines. Furthermore, we develop semi-discrete algorithms that preserve (c) and (d) and fully discrete algorithms that are fourth-order accurate for velocity both in time and in space. These algorithms employ algebraically stable time integrators in a black-box manner and only consist of solving a sequence of linear equations in each time step. Results of numerical tests confirm our analysis.