An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

Abstract

A new implicit stabilized formulation for the numerical solution of the compressible NavierStokes equations is presented. The method is based on the Finite Calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors [24]. In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier-Stokes equations using an implicit scheme is presented.

Adjoint-based adaptive finite element method for the compressible Euler equations using finite calculus

“…More details on the derivation of the proposed stabilized formulation as well as the treatment of the boundary conditions can be found in [44,45]. As discussed in these references, the assumptions β = 0.5 and ℓ = (2Ω e ) 1/2 with Ω e being the element area for 2D problems give suitable results.…”

“…The stabilized formulation of the Euler equations based on the FIC scheme, presented in [44,45], is obtained as…”

“…Within the family of the stabilization techniques, the so called Finite Calculus (FIC) method has been successfully implemented for the stabilization of advective-diffusive transport and incompressible fluid flow problems [38][39][40][41][42][43]. Recently, a FIC-based stabilized formulation for the numerical solution of the compressible Euler and Navier-Stokes equations has been proposed in the context of Galerkin FEM by Kouhi and Oñate [44,45]. Since the systems of equations obtained from both the flow and the adjoint problems contain the same eigenvalues, the stabilization techniques developed for the flow equations can be implemented for solving the adjoint problem as well.…”

“…The system of equations obtained from the discretization of the flow problem is solved implicitly using a damped Newton's method benefiting from the exact Jacobian matrix proposed in [45]. At each iteration step, the inherent linear system resulting from the flow equations is solved with a preconditioned GMRES method.…”

Finite increment calculus (FIC): a framework for deriving enhanced computational methods in mechanics

“…Applications of the FIC/FEM technique to fluid-structure interaction (FSI) problems were reported in [14] (for Eulerian flows) and in [41][42][43][44] (for Lagrangian flows). The FIC/FEM procedure has been also applied to viscous and inviscid compressible flows as reported in [7][8][9].…”