A post-processing technique for stabilizing the discontinuous pressure projection operator in marginally-resolved incompressible inviscid flow. (English) Zbl 1390.76325
Summary: A method for post-processing the velocity after a pressure projection is developed that helps to maintain stability in an under-resolved, inviscid, discontinuous element-based simulation for use in environmental fluid mechanics process studies. The post-processing method is needed because of spurious divergence growth at element interfaces due to the discontinuous nature of the discretization used. This spurious divergence eventually leads to a numerical instability. Previous work has shown that a discontinuous element-local projection onto the space of divergence-free basis functions is capable of stabilizing the projection method, but the discontinuity inherent in this technique may lead to instability in under-resolved simulations. By enforcing inter-element discontinuity and requiring a divergence-free result in the weak sense only, a new post-processing technique is developed that simultaneously improves smoothness and reduces divergence in the pressure-projected velocity field at the same time. When compared against a non-post-processed velocity field, the post-processed velocity field remains stable far longer and exhibits better smoothness and conservation properties.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76B99 | Incompressible inviscid fluids |
Software:
PETScReferences:
| [1] | Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K., Petsc users manual, Tech. Rep. ANL-95/11 - Revision 3.6, (2015), Argonne National Laboratory |
| [2] | Brown, D.; Cortez, R.; Minion, M., Accurate projection methods for the incompressible navierstokes equations, J. Comput. Phys., 168, 464-499, (2001) · Zbl 1153.76339 |
| [3] | Chorin AJ. Numerical solutions of the Navier-Stokes equations. 1968. doi:10.2307/2004575. |
| [4] | Costa, B.; Don, W. S., On the computation of high order pseudospectral derivatives, Appl. Numer. Math., 33, 151-159, (2000) · Zbl 0964.65020 |
| [5] | Deville, M. O.; Fischer, P. F.; Mund, E. H., High-order methods for incompressible fluid flow, (2002), Cambridge Monographs on Applied and Computational Mathematics · Zbl 1007.76001 |
| [6] | Diamessis, P. J.; Domaradzki, J. A.; Hesthaven, J. S., A spectral multidomain penalty method model for the simulation of high Reynolds number localized incompressible stratified turbulence, J. Comput. Phys., 202, 298-322, (2005) · Zbl 1061.76054 |
| [7] | Diamessis, P. J.; Redekopp, L. G., Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers, J. Phys. Oceanogr., 36, 784-812, (2006) |
| [8] | Diamessis, P. J.; Spedding, G. R.; Domaradzki, J. A., Similarity scaling and vorticity structure in high-Reynolds-number stably stratified turbulent wakes, J. Fluid Mech., 671, 52-95, (2011) · Zbl 1225.76167 |
| [9] | Dunphy, M.; Subich, C.; Stastna, M., Spectral methods for internal waves: indistinguishable density profiles and double-humped solitary waves, Nonlinear Process. Geophys., 18, 351-358, (2011) |
| [10] | Escobar-Vargas, J.; Diamessis, P.; Sakai, T., A spectral quadrilateral multidomain penalty method model for high Reynolds number incompressible stratified flows, Int. J. Numer. Methods Fluids, 75, March, 403-425, (2014) · Zbl 1455.65167 |
| [11] | Guermond, J.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195, 6011-6045, (2006) · Zbl 1122.76072 |
| [12] | Guermond, J.-L., Some implementations of projection methods for Navier-Stokes equations, Math. Model. Numer. Anal., 30, 637-667, (1996) · Zbl 0861.76065 |
| [13] | Hesthaven, J. S., A stable penalty method for the compressible Navier-Stokes equations: III. multidimensional domain decomposition schemes, SIAM J. Sci. Comput., 20, 1, 62-93, (1998) · Zbl 0957.76059 |
| [14] | Hesthaven, J. S.; Gottlieb, S.; Gottlieb, D., Spectral methods for time-dependent problems, (2007), Cambridge University Press · Zbl 1111.65093 |
| [15] | Joshi, S. M.; Thomsen, G. N.; Diamessis, P. J., Deflation-accelerated preconditioning of the Poisson-Neumann Schur problem on long domains with a high-order discontinuous element-based collocation method, J. Comput. Phys., 313, 209-232, (2016) · Zbl 1349.65621 |
| [16] | Karniadakis, G. E.; Israeli, M.; Orszag, S.a., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 414-443, (1991) · Zbl 0738.76050 |
| [17] | Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 308-323, (1985) · Zbl 0582.76038 |
| [18] | Kundu, P.; Dowling, D., Fluid mechanics, (2011), Academic Press |
| [19] | Liu, J.-G.; Liu, J.; Pego, R. L., Stable and accurate pressure approximation for unsteady incompressible viscous flow, J. Computat. Phys., 229, 3428-3453, (2010) · Zbl 1307.76029 |
| [20] | Marcus, P. S., Simulation of Taylor-Couette flow. part 1. numerical methods and comparison with experiment, J. Fluid Mech., 146, 45-64, (1984) · Zbl 0561.76037 |
| [21] | Proot, M. M.; Gerritsma, M. I., Application of the least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier-Stokes equations, Numer. Algorithms, 38, 155-172, (2005) · Zbl 1064.76079 |
| [22] | Restelli, M.; Giraldo, F., A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling, SIAM J. Sci. Comput., 31, October, 2231-2257, (2009) · Zbl 1405.65127 |
| [23] | Saad, Y., Iterative methods for sparse linear systems, vol. 3, (2003), Society for Industrial and Applied Mathematics · Zbl 1002.65042 |
| [24] | Sani, R. L.; Shen, J.; Pironneau, O.; Gresho, P. M., Pressure boundary condition for the time-dependent incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 50, August 2004, 673-682, (2006) · Zbl 1155.76322 |
| [25] | Steinmoeller, D.; Stastna, M.; Lamb, K., A short note on the discontinuous Galerkin discretization of the pressure projection operator in incompressible flow, J. Comput. Phys., 251, 480-486, (2013) · Zbl 1349.65488 |
| [26] | Tomboulides, A.; Israeli, M.; Karniadakis, G., Efficient removal of boundary-divergence errors in time-splitting methods, J. Sci. Comput., 4, 3, 291-308, (1989) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
