Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Abstract

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the flow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

Numerical solution of the viscous flows in a network of thin tubes: Equations on the graph

“…We take a cut of the multiply connected geometry in the xy-plane and run a simulation in the interval [0, 0.25ε 2 ] with the Dirichlet boundary conditions for the velocity at the extremities of the three tubes. We use a characteristic Galerkin method, with P 3 − P 2 Taylor-Hood elements in space and BDF2 time integrator (scheme (12) in [2]). It was implemented with FreeFem++ [13].…”

A high-order artificial compressibility method based on Taylor series time-stepping for variable density flow

“…Remark 2.1. To make the extension to variable density to be more in line with Guermond and Minev [21], we chose the advective form of the flux in (5) and (6). To the best of the authors' knowledge, some modifications of the scheme are necessary to achieve an L 2 -estimate.…”

“…One could also consider using a Lagrange-Galerkin approach. The Lagrange-Galerkin method discretizes the total derivative (the convective part of the equations) backward in time along the characteristic curves and has been successfully applied to hyperbolic problems [13] and incompressible flow [6,5]. Investigating if this time-stepping approach is amenable to high-order artificial compressibility would certainly be an interesting research topic.…”

A fully semi-Lagrangian discretization for the 2D incompressible Navier–Stokes equations in the vorticity-streamfunction formulation

“…in which the symbol ∇ ⊥ denotes in compact form the operator defined by (2). Note that the incompressibility condition is ensured by (2) and that the advantage of treating a scalar problem for the vorticity only occurs in two space dimensions. The precise derivation of boundary conditions for the vorticitystreamfunction formulation is not trivial, see e.g.…”

“…on a bounded domain Ω ⊂ R 2 , with proper initial and boundary conditions. SL techniques of both the forms outlined above have been proposed for this problem, we refer here for example to the papers [16], [29], [31], [30] and the review in [2], in which the application of the SL method is restricted to the advective part, and to [1], [20], which present and analyze a fully SL approach based on a stochastic framework. In particular, these latter works present a theoretical analysis for the time-discrete scheme, although with limited numerical validation.…”