High-Order Time Stepping for the Incompressible Navier--Stokes Equations

Abstract

This paper introduces a high-order time stepping technique for solving the incompressible Navier--Stokes equations which, unlike coupled techniques, does not require solving a saddle point problem at each time step and, unlike projection methods, does not produce splitting errors and spurious boundary layers. The technique is a generalization of the artificial compressibility method; it is unconditionally stable (for the unsteady Stokes equations), can reach any order in time, and uncouples the velocity and the pressure. The condition number of the linear systems associated with the fully discrete vector-valued problems to be solved at each time step scales like $O(\tau h^{-2})$, where $\tau$ is the time step and $h$ is the spatial grid size. No Poisson problem or other second-order elliptic problem has to be solved for the pressure corrections. Unlike projection methods, optimal convergence is observed numerically with Dirichlet and mixed Dirichlet/Neumann boundary conditions.