Steady Incremental Viscosity Splitting Method for solving the stationary Navier-Stokes equation

TL;DR

This paper introduces a new iterative method for efficiently solving the stationary Navier-Stokes equations by adapting an incremental viscosity splitting approach, with proven convergence and demonstrated numerical efficiency.

Contribution

The paper presents a novel steady-state viscosity splitting scheme with proven convergence and efficiency for solving the stationary Navier-Stokes equations.

Findings

The method requires solving an elliptic PDE and an SPD system per iteration.

The scheme is proven to be bounded and geometrically convergent.

Numerical tests confirm the efficiency of the algorithm.

Abstract

We develop a novel and efficient iterative scheme for solving incompressible steady Navier-Stokes equations. The method is an adaptation of the Incremental Viscosity Splitting approximation for unsteady flows to steady equations. At each nonlinear iteration, the scheme requires solving an elliptic PDE for the velocity variable and a system with an SPD matrix for the pressure variable, which remains the same across all nonlinear iterations. The method can also be interpreted as an algebraic splitting approach. We prove boundedness and geometric convergence. Numerical tests illustrate the efficiency of the proposed algorithm.