Abstract
We develop a semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1-form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of the scheme enjoys excellent stability properties in the vanishing viscosity limit and is applicable to inviscid incompressible Euler flows. We obtain second-order convergence and conservation of energy is achieved through a Lagrange multiplier.
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Mitrea, M., Monniaux, S.: The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains. Differ. Integral Equ. 22(3/4), 339–356 (2009). https://doi.org/10.57262/die/1356019778
Temam, R., Ziane, M.: Navier-Stokes equations in three-dimensional thin domains with various boundary conditions. Adv. Differ. Equ. 1(4), 499–546 (1996). https://doi.org/ade/1366896027
Arnold, V.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications á l’hydrodynamique des fluides parfaits. Annales de L’Institut Fourier 16, 319–361 (1966)
Natale, A., Cotter, C.J.: A variational H(div) finite-element discretization approach for perfect incompressible fluids. IMA J. Numer. Anal. 38(3), 1388–1419 (2018). https://doi.org/10.1093/imanum/drx033
Chorin, A.J., Marsden, J.E.: A mathematical introduction to fluid mechanics. Springer (1993). https://doi.org/10.1007/978-1-4612-0883-9
De Rosa, L.: On the Helicity conservation for the incompressible Euler equations. Proc. Am. Math. Soc. 148(7), 2969–2979 (2020). https://doi.org/10.1090/proc/14952
Isett, P.: A proof of Onsager’s conjecture. Ann. Math. 188(3), 871–963 (2018). https://doi.org/10.4007/annals.2018.188.3.4
Bercovier, M., Pironneau, O.: Characteristics and the finite element method. Finite Element Flow Analysis, pp. 67–73 (1982)
Bercovier, M., Pironneau, O., Sastri, V.: Finite elements and characteristics for some parabolic-hyperbolic problems. Appl. Math. Model. 7(2), 89–96 (1983)
Douglas, J., Jr., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19(5), 871–885 (1982)
Ewing, R.E., Russell, T.F., Wheeler, M.F.: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Eng. 47(1–2), 73–92 (1984)
Hasbani, Y., Livne, E., Bercovier, M.: Finite elements and characteristics applied to advection-diffusion equations. Comput. Fluids 11(2), 71–83 (1983)
Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38(3), 309–332 (1982). https://doi.org/10.1007/BF01396435
Russell, T.F.: Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media. SIAM J. Numer. Anal. 22(5), 970–1013 (1985)
Süli, E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53(4), 459–483 (1988). https://doi.org/10.1007/BF01396329
Bause, M., Knabner, P.: Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems. SIAM J. Numer. Anal. 39(6), 1954–1984 (2002). https://doi.org/10.1137/S0036142900367478
Bermejo, R., Saavedra, L.: Modified Lagrange-Galerkin methods of first and second order in time for convection-diffusion problems. Numer. Math. 120(4) (2012). https://doi.org/10.1007/s00211-011-0418-8
Wang, H., Wang, K.: Uniform estimates of an eulerian-lagrangian method for time-dependent convection-diffusion equations in multiple space dimensions. SIAM J. Numer. Anal. 48(4), 1444–1473 (2010). https://doi.org/10.1137/070682952
Heumann, H., Hiptmair, R.: Convergence of lowest order semi-lagrangian schemes. Found. Comput. Math. 13(2), 187–220 (2013). https://doi.org/10.1007/s10208-012-9139-3
Bermejo, R., Saavedra, L.: Lagrange-Galerkin methods for the incompressible Navier-Stokes equations: a review. Commun. Appl. Ind. Math. 7(3), 26–55 (2016). https://doi.org/10.1515/caim-2016-0021
Heumann, H., Hiptmair, R., Li, K., Xu, J.: Fully discrete semi-Lagrangian methods for advection of differential forms. BIT Numer. Math. 52(4), 981–1007 (2012). https://doi.org/10.1007/s10543-012-0382-4
Heumann, H., Hiptmair, R.: Eulerian and semi-lagrangian methods for convection-diffusion for differential forms, pp. 1–26 (2011). https://doi.org/10.3929/ethz-a-006506738
Boukir, K., Maday, Y., Métivet, B.: A high order characteristics method for the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 116(1–4), 211–218 (1994). https://doi.org/10.1016/S0045-7825(94)80025-1
Buscaglia, G.C., Dari, E.A.: Implementation of the Lagrange-Galerkin method for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 15(1), 23–36 (1992). https://doi.org/10.1002/fld.1650150103
Minev, P.D., Ross Ethier, C.: A characteristic/finite element algorithm for the 3-D Navier-Stokes equations using unstructured grids. Comput. Methods Appl. Mech. Eng. 178(1–2), 39–50 (1999). https://doi.org/10.1016/S0045-7825(99)00003-1
El-Amrani, M., Seaid, M.: An L2-projection for the Galerkin-characteristic solution of incompressible flows. SIAM J. Sci. Comput. 33(6), 3110–3131 (2011). https://doi.org/10.1137/100805790
Bermejo, R., Saavedra, L.: Modified lagrange-galerkin methods to integrate time dependent incompressible navier-stokes equations. SIAM J. Sci. Comput. 37(6), 779–803 (2015). https://doi.org/10.1137/140973967
Bermejo, R., Galán Del Sastre, P., Saavedra, L.: A second order in time modified lagrange-galerkin finite element method for the incompressible navier-stokes equations. SIAM J. Numer. Anal. 50(6), 3084–3109 (2012)
Morton, K.W., Priestley, A., Suli, E.: Stability of the Lagrange-Galerkin method with non-exact Integration. Math. Model. Numer. Anal. 22(4), 625–653 (1988)
Patera, A.T.: A spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54(3), 468–488 (1984). https://doi.org/10.1016/0021-9991(84)90128-1
Karniadakis, G., Sherwin, S.: Spectral/hp element methods for computational fluid dynamics. Oxford University Press (2005). https://doi.org/10.1093/acprof:oso/9780198528692.001.0001
Xiu, D., Karniadakis, G.E.: A Semi-lagrangian high-order method for navier-stokes equations. J. Comput. Phys. 172(2), 658–684 (2001). https://doi.org/10.1006/jcph.2001.6847
Xiu, D., Sherwin, S.J., Dong, S., Karniadakis, G.E.: Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows. J. Sci. Comput. 25(1), 323–346 (2005). https://doi.org/10.1007/s10915-004-4647-1
Bonaventura, L., Ferretti, R., Rocchi, L.: A fully semi-Lagrangian discretization for the 2D incompressible Navier-Stokes equations in the vorticity-streamfunction formulation. Appl. Math. Comput. 323, 132–144 (2018). https://doi.org/10.1016/j.amc.2017.11.030
Celledoni, E., Kometa, B.K., Verdier, O.: High order semi-lagrangian methods for the incompressible navier-stokes equations. J. Sci. Comput. 66(1), 91–115 (2016). https://doi.org/10.1007/s10915-015-0015-6
Celledoni, E., Marthinsen, A., Owren, B.: Commutator-free Lie group methods. Futur. Gener. Comput. Syst. 19(3), 341–352 (2003). https://doi.org/10.1016/S0167-739X(02)00161-9
Arnold, D.N.: Finite Element Exterior Calculus. Society for Industrial and Applied Mathematics, Philadelphia, PA (2018). https://doi.org/10.1137/1.9781611975543
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006). https://doi.org/10.1017/S0962492906210018
Rapetti, F., Bossavit, A.: Whitney forms of higher degree. SIAM J. Numer. Anal. 47(3), 2369–2386 (2009). https://doi.org/10.1137/070705489
Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica 11, 237–339 (2002). https://doi.org/10.1017/S0962492902000041
Süli, E., Mayers, D.F.: An introduction to numerical analysis. Cambridge University Press (2003). https://doi.org/10.1017/CBO9780511801181
Vuik, C., Vermolen, F.J., Gijzen, M.B., Vuik, M.J.: Numerical methods for ordinary differential equations. Delft Academic Press (2015). https://doi.org/10.1017/s10915-004-4647-1
Anderson, R., Andrej, J., Barker, A., Bramwell, J., Camier, J.S., Cerveny, J., Dobrev, V., Dudouit, Y., Fisher, A., Kolev, T., Pazner, W., Stowell, M., Tomov, V., Akkerman, I., Dahm, J., Medina, D., Zampini, S.: MFEM: A modular finite element methods library. Comput. Math. Appl. 81 (2021). https://doi.org/10.1016/j.camwa.2020.06.009
Taylor, G.I., Green, A.E.: Mechanism of the production of small eddies from large ones. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences 158(895), 499–521 (1937). https://doi.org/10.1098/rspa.1937.0036
Popinet, S.: The gerris flow solver (2007). https://gfs.sourceforge.net/wiki/index.php/Main_/Page
Popinet, S.: Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190(2), 572–600 (2003). https://doi.org/10.1016/S0021-9991(03)00298-5
Geuzaine, C., Remacle, J.-F.: Gmsh: A 3-D finite element mesh generator with builtin pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009). https://doi.org/10.1002/nme.2579
Acknowledgements
The authors of this article are greatly indebted to Prof. Alain Bossavit for his many seminal contributions to finite element exterior calculus even before that term was coined. His work has deeply influenced their research and, in particular, his discovery of the role of small simplices as redundant degrees of freedom has paved the way for the results reported in the present paper.
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Appendix A: Two formulations of the momentum equation
Appendix A: Two formulations of the momentum equation
Consider the momentum equation in ()
Note that we have by standard vector calculus identities
where we can use \(\nabla \cdot \varvec{u}=0\) to obtain
This allows us to rewrite the momentum equation as
Using the gradient of the dot-product, we find
This identity allows us to rewrite the momentum equation to
From [19, 40], we obtain the identity
where \(\omega \) is the differential 1-form such that . Since the material derivative for this 1-form is
we find that the momentum equation can be written as
where \(\tilde{p} = -\frac{1}{2}\varvec{u}\cdot \varvec{u}+p\).
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Tonnon, W., Hiptmair, R. Semi-Lagrangian finite element exterior calculus for incompressible flows. Adv Comput Math 50, 11 (2024). https://doi.org/10.1007/s10444-023-10092-6
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DOI: https://doi.org/10.1007/s10444-023-10092-6