Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p varia- tional methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable (18), when h- or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic vis- cosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used. Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with Courant- Friedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial or- der or a mall enough mesh spacing is selected. AMS subject classifications: 76D05, 35Q30, 65N30

[1]  Esteban Ferrer,et al.  A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier-Stokes equations , 2011 .

[2]  Jie Shen,et al.  PSEUDO-COMPRESSIBILITY METHODS FOR THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 2007 .

[3]  R. Temam.  Une méthode d'approximation de la solution des équations de Navier-Stokes , 1968 .

[4]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[5]  Jean-Luc Guermond,et al.  International Journal for Numerical Methods in Fluids on Stability and Convergence of Projection Methods Based on Pressure Poisson Equation , 2022 .

[6]  F. B. Ellerby,et al.  Numerical solutions of partial differential equations by the finite element method , by C. Johnson. Pp 278. £40 (hardback), £15 (paperback). 1988. ISBN 0-521-34514-6, 34758-0 (Cambridge University Press) , 1989, The Mathematical Gazette.

[7]  P. Fischer,et al.  High-Order Methods for Incompressible Fluid Flow , 2002 .

[8]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[9]  C. Ross Ethier,et al.  A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations , 2007, J. Comput. Phys..

[10]  Marek Stastna,et al.  A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow , 2013, J. Comput. Phys..

[11]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[12]  Einar M. Rønquist,et al.  An Operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow , 1990 .

[13]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[14]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[15]  R. Rannacher.  On chorin's projection method for the incompressible navier-stokes equations , 1992 .

[16]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[17]  Jie Shen,et al.  Velocity-Correction Projection Methods for Incompressible Flows , 2003, SIAM J. Numer. Anal..

[18]  Guido Kanschat,et al.  An Equal-Order DG Method for the Incompressible Navier-Stokes Equations , 2009, J. Sci. Comput..

[19]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 2014 .

[20]  R. Codina,et al.  Algebraic Pressure Segregation Methods for the Incompressible Navier-Stokes Equations , 2007 .

[21]  L. D. Marini,et al.  Stabilization mechanisms in discontinuous Galerkin finite element methods , 2006 .

[22]  R. Hartmann.  Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element methods , 2008 .

[23]  Ning Hu,et al.  Bounds for eigenvalues and condition numbers in the p-version of the finite element method , 1998, Math. Comput..

[24]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[25]  F. Brezzi,et al.  A discourse on the stability conditions for mixed finite element formulations , 1990 .

[26]  S. Orszag,et al.  Boundary conditions for incompressible flows , 1986 .

[27]  N. Parolini,et al.  Role of the LBB condition in weak spectral projection methods: 405 , 2001 .

[28]  Richard H. J. Willden.  A high order Discontinuous Galerkin - Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes for simulating cross-flow turbines , 2012 .

[29]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[30]  R. Codina.  Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows , 2001 .

[31]  M. G. Duffy,et al.  Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex , 1982 .

[32]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[33]  B. Rivière,et al.  Estimation of penalty parameters for symmetric interior penalty Galerkin methods , 2007 .

[34]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[35]  Anthony T. Patera,et al.  Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations , 1993, SIAM J. Sci. Comput..

[36]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 1972 .