An Augmented Lagrangian Preconditioner for Navier--Stokes Equations with Runge--Kutta in Time

TL;DR

This paper introduces an augmented Lagrangian preconditioner combined with Runge--Kutta time integration for efficiently solving the nonlinear saddle-point systems arising from discretized Navier--Stokes equations, demonstrating robustness and efficiency.

Contribution

It develops a novel augmented Lagrangian preconditioner tailored for Runge--Kutta discretized Navier--Stokes equations, improving solver robustness and efficiency.

Findings

Preconditioner is effective across various viscosities and mesh sizes.

Numerical results show robustness and efficiency of the method.

Inexact application with multigrid is feasible and beneficial.

Abstract

We consider a Runge--Kutta method for the numerical time integration of the nonstationary incompressible Navier--Stokes equations. This yields a sequence of nonlinear problems to be solved for the stages of the Runge--Kutta method. The resulting nonlinear system of differential equations is discretized using a finite element method. To compute a numerical approximation of the stages at each time step, we employ Newton's method, which requires the solution of a large and sparse generalized saddle-point problem at each nonlinear iteration. We devise an augmented Lagrangian preconditioner within the flexible GMRES method for solving the Newton systems at each time step. The preconditioner can be applied inexactly with the help of a multigrid routine. We present numerical evidence of the robustness and efficiency of the proposed strategy for different values of the viscosity, mesh size, time step, and number of stages of the Runge--Kutta method.