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.

Abstract

Abstract

Solution methods for systems of ordinary differential equations with variable step size base their choice for the step size on (an estimate of) the local truncation error. Significant savings in computational effort can be obtained as the step size is only refined when necessary. We first summarize the information that is needed for a variable step size BDF2 method in a way that is directly suitable for implementation in an existing code. We then propose a solution verification procedure that does not require direct control of the step size to carry out refinement studies and apply it to a textbook example. Finally, we explore the use of the variable step size BDF2 method for time integration of the NavierStokes equations in the CFD package REFRESCO and find plausible results for vortex shedding in the wake of a cylinder, but no major savings for this particular case.

Abstract