# High-order time stepping for the Navier-Stokes equations with minimal computational complexity

@article{Guermond2017HighorderTS, title={High-order time stepping for the Navier-Stokes equations with minimal computational complexity}, author={Jean-Luc Guermond and Peter Dimitrov Minev}, journal={J. Comput. Appl. Math.}, year={2017}, volume={310}, pages={92-103} }

In this paper we present extensions of the schemes proposed in Guermond and Minev (2015) that lead to a decoupling of the velocity components in the momentum equation. The new schemes reduce the solution of the incompressible Navier-Stokes equations to a set of classical scalar parabolic problems for each Cartesian component of the velocity. The pressure is explicitly recovered after the velocity is computed.

#### 14 Citations

Numerical Analysis of a BDF2 Modular Grad–Div Stabilization Method for the Navier–Stokes Equations

- Mathematics, Computer Science
- J. Sci. Comput.
- 2020

A second-order accurate modular algorithm is presented for a standard BDF2 code for the Navier–Stokes equations (NSE). The algorithm exhibits resistance to solver breakdown and increased… Expand

High-order Methods for a Pressure Poisson Equation Reformulation of the Navier-Stokes Equations with Electric Boundary Conditions

- Mathematics, Computer Science
- ArXiv
- 2020

It is studied to what extent high-order methods for the NSE can be obtained from a specific PPE reformulation with electric boundary conditions (EBC); implicit-explicit time-stepping is used to decouple the pressure solve from the velocity update, while avoiding a parabolic time-step restriction. Expand

High-order finite element methods for a pressure Poisson equation reformulation of the Navier–Stokes equations with electric boundary conditions

- Mathematics
- 2021

Abstract Pressure Poisson equation (PPE) reformulations of the incompressible Navier–Stokes equations (NSE) replace the incompressibility constraint by a Poisson equation for the pressure and a… Expand

Splitting schemes for unsteady problems involving the grad-div operator

- Mathematics, Computer Science
- ArXiv
- 2016

Various possibilities to decouple the equations for the different components that result in unconditionally stable schemes are discussed. Expand

A direction splitting scheme for Navier-Stokes-Boussinesq system in spherical shell geometries

- Physics, Mathematics
- 2019

This paper introduces a formally second-order direction-splitting method for solving the incompressible Navier-Stokes-Boussinesq system in a spherical shell region based on second order finite differences on the Marker-And-Cell stencil. Expand

An efficient and modular grad–div stabilization

- Computer Science, Mathematics
- 2018

Two modular grad–div algorithms for calculating solutions to the Navier–Stokes equations (NSE) are presented, adding to an NSE code a minimally intrusive module that implements grad-div stabilization. Expand

A conservative, second order, unconditionally stable artificial compression method

- Computer Science
- 2017

The method has second order consistency error and is unconditionally, long time, energy stable for the velocity and, weighted by the timestep, for the pressure and requires no artificial pressure boundary conditions. Expand

Computationally efficient modular nonlinear filter stabilization for high Reynolds number flows

- Mathematics, Computer Science
- Adv. Comput. Math.
- 2018

This work proposes a computationally efficient version of the filtering step that only requires the assembly once, and the solution of two symmetric, positive definite systems at each time step, and test a new indicator function based on the entropy viscosity model. Expand

Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part II: Comparisons of local error estimation and step-selection strategies for nonlinear Schrödinger and wave equations

- Physics, Mathematics
- Comput. Phys. Commun.
- 2019

For nonlinear wave equations, the enhanced computational stability ensuing from adaptive step selection strategies close to the border mandated by the CFL condition is demonstrated. Expand

Doubly-adaptive artificial compression methods for incompressible flow

- Computer Science, Mathematics
- J. Num. Math.
- 2020

Adaptive artificial compression methods in which the time-step and artificial compression parameter ε are independently adapted are presented, and the computational, cognitive, and space complexities are negligibly greater than that of the simplest, first-order, constant ε, constant k artificial compression method. Expand

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