doi: 10.3934/jcd.2019001

Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations

1. 

Imperial College London, Department of Mathematics, 180 Queen's Gate, London SW7 2AZ, United Kingdomand

2. 

École Normale Supérieure, Laboratoire de Météorologie Dynamique, 24 Rue Lhomond, 75005 Paris, France

3. 

CNRS/École Normale Supérieure, Laboratoire de Météorologie Dynamique, 24 Rue Lhomond, 75005 Paris, France

Received  December 2017 Revised  September 2018 Published  December 2018

This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the scheme numerically for the rotating shallow water equations. In particular, we investigate whether the scheme conserves stationary solutions, represents well the nonlinear dynamics, and approximates well the frequency relations of the continuous equations, while preserving conservation laws such as mass and total energy.

Citation: Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, doi: 10.3934/jcd.2019001
References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, Grenoble, 16 (1966), 319-361. doi: 10.5802/aif.233.

[2]

R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, part Ⅰ: Grids with superconvergence, SIAM Journal on Numerical Analysis, 41 (2003), 2294-2312. doi: 10.1137/S003614290139874X.

[3]

W. Bauer, Toward Goal-Oriented R-adaptive Models in Geophysical Fluid Dynamics using a Generalized Discretization Approach, Ph.D thesis, Department of Geosciences, University of Hamburg, 2013.

[4]

W. BauerM. BaumannL. ScheckA. GassmannV. Heuveline and S. C. Jones, Simulation of tropical-cyclone-like vortices in shallow-water icon-hex using goal-oriented r-adaptivity, Theoretical and Computational Fluid Dynamics, 28 (2014), 107-128. doi: 10.1007/s00162-013-0303-4.

[5]

W. Bauer, A new hierarchically-structured $n$-dimensional covariant form of rotating equations of geophysical fluid dynamics, GEM - International Journal on Geomathematics, 7 (2016), 31-101. doi: 10.1007/s13137-015-0074-8.

[6]

W. Bauer and F. Gay-Balmaz, Variational integrators for the anelastic and pseudo-incompressible flows, preprint, 2017, arXiv: 1701.06448.

[7]

A. M. Bloch, Nonholonomic Mechanics and Control, Volume 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2003. With the collaboration of J. Baillieul, P. Crouch and J. E. Marsden, and with scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov. doi: 10.1007/b97376.

[8]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups. Part Ⅰ: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.

[9]

R. Brecht, W. Bauer, A. Bihlo, F. Gay-Balmaz and S. MacLachlan, Variational integrator for the rotating shallow-water equations on the sphere, preprint, 2018, arXiv: 1808.10507.

[10]

W. Cao, Superconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes, Math. Comput., 84 (2015), 89-117. doi: 10.1090/S0025-5718-2014-02846-9.

[11] M. J. P. Cullen, A Mathematical Theory of Large-scale Atmosphere/ocean Flow, Imperial College Press, London, 2006. doi: 10.1142/p375.
[12]

F. Demoures, F. Gay-Balmaz and T. S. Ratiu, Multisymplectic variational integrators for nonsmooth Lagrangian continuum mechanics, Forum of Mathematics, Sigma, 4 (2016), e19, 54 pp. doi: 10.1017/fms.2016.17.

[13]

F. DemouresF. Gay-BalmazM. Kobilarov and T. S. Ratiu, Multisymplectic Lie group variational integrators for a geometrically exact beam in $ \mathbb{R} ^3 $, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 3492-3512. doi: 10.1016/j.cnsns.2014.02.032.

[14]

M. DesbrunE. GawlikF. Gay-Balmaz and V. Zeitlin, Variational discretization for rotating stratified fluids, Disc. Cont. Dyn. Syst. Series A, 34 (2014), 479-511. doi: 10.3934/dcds.2014.34.477.

[15]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, 2004. doi: 10.1007/978-1-4757-4355-5.

[16]

E. GawlikP. MullenD. PavlovJ. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760. doi: 10.1016/j.physd.2011.07.011.

[17]

M. Giorgetta, T. Hundertmark, P. Korn, S. Reich and M. Restelli, Conservative space and time regularizations for the icon model, Technical report, Berichte zur Erdsystemforschung, Report 67, MPI for Meteorology, Hamburg, 2009.

[18]

F. Gay-Balmaz and V. Putkaradze, Variational discretizations for the dynamics of fluid-conveying flexible tubes, C. R. Mécanique, 344 (2016), 769-775. doi: 10.1016/j.crme.2016.08.004.

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, 2006.

[20]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[21]

A. LewJ. E. MarsdenM. Ortiz and M. West, Asynchronous variational integrators, Arch. Rat. Mech. Anal., 167 (2003), 85-146. doi: 10.1007/s00205-002-0212-y.

[22]

Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids, Mathematics of Computation, 77 (2008), 1253-1268. doi: 10.1090/S0025-5718-08-02051-6.

[23]

R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058.

[24]

B. Liu, G. Mason, J. Hodgson, Y. Tong and M. Desbrun, Model-reduced variational fluid simulation, ACM Trans. Graph. (SIG Asia), 34 (2015), Art. 244. doi: 10.1145/2816795.2818130.

[25]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[26]

J. E. MarsdenG. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505.

[27]

D. PavlovP. MullenY. TongE. KansoJ. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D, 240 (2010), 443-458. doi: 10.1016/j.physd.2010.10.012.

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, New York, 1979.

[29]

S. Reich, Linearly implicit time stepping methods for numerical weather prediction, BIT Numerical Mathematics, 46 (2006), 607-616. doi: 10.1007/s10543-006-0065-0.

[30]

S. ReichN. Wood and A. Staniforth, Semi-implicit methods, nonlinear balance, and regularized equations, Atmospheric Science Letters, 8 (2007), 1-6. doi: 10.1002/asl.142.

[31]

T. D. RinglerJ. ThuburnJ. B. Klemp and W. C. Skamarock, A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids, J. Comput. Phys., 229 (2010), 3065-3090. doi: 10.1016/j.jcp.2009.12.007.

[32]

A. Staniforth and A. A. White, Some exact solutions of geophysical fluid-dynamics equations for testing models in spherical and plane geometry, Q. J. R. Meteorol. Soc., 133 (2007), 1605-1614. doi: 10.1002/qj.122.

[33]

A. StaniforthN. Wood and S. Reich, A time-staggered semi-lagrangian discretization of the rotating shallow-water equations, Quarterly Journal of the Royal Meteorological Society, 132 (2006), 3107-3116. doi: 10.1256/qj.06.30.

[34]

A. Stegner and D. Dritschel, A numerical investigation of the stability of isolated shallow-water vortices, J. Phys. Ocean., 30 (2000), 2562-2573. doi: 10.1175/1520-0485(2000)030<2562:ANIOTS>2.0.CO;2.

[35]

J. ThuburnT. D. RinglerW. C. Skamarock and J. B. Klemp, Numerical representation of geostrophic modes on arbitrarily structured C-grids, J. Comput. Phys., 228 (2009), 8321-8335. doi: 10.1016/j.jcp.2009.08.006.

[36]

J. Thuburn and C. J. Cotter, A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, Journal of Computational Physics, 290 (2015), 274-297. doi: 10.1016/j.jcp.2015.02.045.

[37]

D. L. WilliamsonJ. B. DrakeJ. J. HackR. Jakob and P. N. Swarztrauber, A standard test set for numerical approximations to the shallow-water equations in spherical geometry, J. Comput. Phys., 102 (1992), 221-224. doi: 10.1016/S0021-9991(05)80016-6.

[38]

V. Zeitlin (Ed.), Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances, Elsevier, New York, 2007.

show all references

References:
[1]

V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier, Grenoble, 16 (1966), 319-361. doi: 10.5802/aif.233.

[2]

R. E. Bank and J. Xu, Asymptotically exact a posteriori error estimators, part Ⅰ: Grids with superconvergence, SIAM Journal on Numerical Analysis, 41 (2003), 2294-2312. doi: 10.1137/S003614290139874X.

[3]

W. Bauer, Toward Goal-Oriented R-adaptive Models in Geophysical Fluid Dynamics using a Generalized Discretization Approach, Ph.D thesis, Department of Geosciences, University of Hamburg, 2013.

[4]

W. BauerM. BaumannL. ScheckA. GassmannV. Heuveline and S. C. Jones, Simulation of tropical-cyclone-like vortices in shallow-water icon-hex using goal-oriented r-adaptivity, Theoretical and Computational Fluid Dynamics, 28 (2014), 107-128. doi: 10.1007/s00162-013-0303-4.

[5]

W. Bauer, A new hierarchically-structured $n$-dimensional covariant form of rotating equations of geophysical fluid dynamics, GEM - International Journal on Geomathematics, 7 (2016), 31-101. doi: 10.1007/s13137-015-0074-8.

[6]

W. Bauer and F. Gay-Balmaz, Variational integrators for the anelastic and pseudo-incompressible flows, preprint, 2017, arXiv: 1701.06448.

[7]

A. M. Bloch, Nonholonomic Mechanics and Control, Volume 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2003. With the collaboration of J. Baillieul, P. Crouch and J. E. Marsden, and with scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov. doi: 10.1007/b97376.

[8]

N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups. Part Ⅰ: Introduction and structure-preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4.

[9]

R. Brecht, W. Bauer, A. Bihlo, F. Gay-Balmaz and S. MacLachlan, Variational integrator for the rotating shallow-water equations on the sphere, preprint, 2018, arXiv: 1808.10507.

[10]

W. Cao, Superconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes, Math. Comput., 84 (2015), 89-117. doi: 10.1090/S0025-5718-2014-02846-9.

[11] M. J. P. Cullen, A Mathematical Theory of Large-scale Atmosphere/ocean Flow, Imperial College Press, London, 2006. doi: 10.1142/p375.
[12]

F. Demoures, F. Gay-Balmaz and T. S. Ratiu, Multisymplectic variational integrators for nonsmooth Lagrangian continuum mechanics, Forum of Mathematics, Sigma, 4 (2016), e19, 54 pp. doi: 10.1017/fms.2016.17.

[13]

F. DemouresF. Gay-BalmazM. Kobilarov and T. S. Ratiu, Multisymplectic Lie group variational integrators for a geometrically exact beam in $ \mathbb{R} ^3 $, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 3492-3512. doi: 10.1016/j.cnsns.2014.02.032.

[14]

M. DesbrunE. GawlikF. Gay-Balmaz and V. Zeitlin, Variational discretization for rotating stratified fluids, Disc. Cont. Dyn. Syst. Series A, 34 (2014), 479-511. doi: 10.3934/dcds.2014.34.477.

[15]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, 2004. doi: 10.1007/978-1-4757-4355-5.

[16]

E. GawlikP. MullenD. PavlovJ. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760. doi: 10.1016/j.physd.2011.07.011.

[17]

M. Giorgetta, T. Hundertmark, P. Korn, S. Reich and M. Restelli, Conservative space and time regularizations for the icon model, Technical report, Berichte zur Erdsystemforschung, Report 67, MPI for Meteorology, Hamburg, 2009.

[18]

F. Gay-Balmaz and V. Putkaradze, Variational discretizations for the dynamics of fluid-conveying flexible tubes, C. R. Mécanique, 344 (2016), 769-775. doi: 10.1016/j.crme.2016.08.004.

[19]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31, Springer-Verlag, 2006.

[20]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721.

[21]

A. LewJ. E. MarsdenM. Ortiz and M. West, Asynchronous variational integrators, Arch. Rat. Mech. Anal., 167 (2003), 85-146. doi: 10.1007/s00205-002-0212-y.

[22]

Y. Huang and J. Xu, Superconvergence of quadratic finite elements on mildly structured grids, Mathematics of Computation, 77 (2008), 1253-1268. doi: 10.1090/S0025-5718-08-02051-6.

[23]

R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058.

[24]

B. Liu, G. Mason, J. Hodgson, Y. Tong and M. Desbrun, Model-reduced variational fluid simulation, ACM Trans. Graph. (SIG Asia), 34 (2015), Art. 244. doi: 10.1145/2816795.2818130.

[25]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X.

[26]

J. E. MarsdenG. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505.

[27]

D. PavlovP. MullenY. TongE. KansoJ. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D, 240 (2010), 443-458. doi: 10.1016/j.physd.2010.10.012.

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, New York, 1979.

[29]

S. Reich, Linearly implicit time stepping methods for numerical weather prediction, BIT Numerical Mathematics, 46 (2006), 607-616. doi: 10.1007/s10543-006-0065-0.

[30]

S. ReichN. Wood and A. Staniforth, Semi-implicit methods, nonlinear balance, and regularized equations, Atmospheric Science Letters, 8 (2007), 1-6. doi: 10.1002/asl.142.

[31]

T. D. RinglerJ. ThuburnJ. B. Klemp and W. C. Skamarock, A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids, J. Comput. Phys., 229 (2010), 3065-3090. doi: 10.1016/j.jcp.2009.12.007.

[32]

A. Staniforth and A. A. White, Some exact solutions of geophysical fluid-dynamics equations for testing models in spherical and plane geometry, Q. J. R. Meteorol. Soc., 133 (2007), 1605-1614. doi: 10.1002/qj.122.

[33]

A. StaniforthN. Wood and S. Reich, A time-staggered semi-lagrangian discretization of the rotating shallow-water equations, Quarterly Journal of the Royal Meteorological Society, 132 (2006), 3107-3116. doi: 10.1256/qj.06.30.

[34]

A. Stegner and D. Dritschel, A numerical investigation of the stability of isolated shallow-water vortices, J. Phys. Ocean., 30 (2000), 2562-2573. doi: 10.1175/1520-0485(2000)030<2562:ANIOTS>2.0.CO;2.

[35]

J. ThuburnT. D. RinglerW. C. Skamarock and J. B. Klemp, Numerical representation of geostrophic modes on arbitrarily structured C-grids, J. Comput. Phys., 228 (2009), 8321-8335. doi: 10.1016/j.jcp.2009.08.006.

[36]

J. Thuburn and C. J. Cotter, A primal-dual mimetic finite element scheme for the rotating shallow water equations on polygonal spherical meshes, Journal of Computational Physics, 290 (2015), 274-297. doi: 10.1016/j.jcp.2015.02.045.

[37]

D. L. WilliamsonJ. B. DrakeJ. J. HackR. Jakob and P. N. Swarztrauber, A standard test set for numerical approximations to the shallow-water equations in spherical geometry, J. Comput. Phys., 102 (1992), 221-224. doi: 10.1016/S0021-9991(05)80016-6.

[38]

V. Zeitlin (Ed.), Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances, Elsevier, New York, 2007.

Figure 1.  Notation and indexing conventions for the 2D simplicial mesh.
Figure 2.  Regular mesh with equilateral triangles and irregular mesh with central refinement region, both with $ 2 \cdot 32^2 $ triangular grid cells.
Figure 3.  Left: contour lines of the bottom topography $ B(x, y) $ on the computational domain. Right: maximum errors in surface elevation at rest relative to $ H_0 = 750 $m for regular (upper right) and irregular (lower right) meshes.
Figure 4.  Frequency spectra of the disturbed lake at rest after 10 days for parameters $ f = 5.31 $ $ \rm days^{-1} $, $ H_0 = 750 $m (left) or $ f = 6.903 $ $ \rm days^{-1} $, $ H_0 = 1267.5 $m (right) determined on an irregular mesh with $ 2 \cdot 64^2 $ cells. The frequency spectra determined on regular meshes looks very similar (not shown).
Figure 5.  Isolated vortex test case: fluid depth $ D(x, y) $ at initial time $ t = 0 $ (left) and at $ t = 100 $days on a regular (center) and an irregular (right) mesh with $ 2 \cdot 64^2 $ triangular cells. Contours between $ 0.698 {\rm km} $ and $ 0.752 {\rm km} $ with interval of $ 0.003 {\rm km} $.
Figure 6.  Isolated vortex test case: relative potential vorticity $ q_{\rm rel}(x, y) $ at initial time $ t = 0 $ (left) and at $ t = 100 $days on a regular (center) and an irregular (right) mesh with $ 2 \cdot 64^2 $ triangular cells. Contours between $ -1.5 {\rm days^{-1}km^{-1}} $ and $ 12.5 {\rm days^{-1}km^{-1}} $ with interval of $ 1 {\rm days^{-1}km^{-1}} $.
Figure 8.  Isolated vortex test case: $ L_2 $ and $ L_\infty $ error values of numerical solutions for $ D $ and $ q_{\rm rel} $ after 1 day as a function of grid resolution for a fluid in semi-geostrophic (left), quasi-geostrophic (middle), and incompressible (right) regime for regular and irregular meshes.
Figure 7.  Isolated vortex test case: relative errors of total energy $ E(t) $ on meshes with $ 2\cdot 64^2 $ cells (upper row) and with $ 2\cdot 32^2 $ cells (lower row) for a fluid in semi-geostrophic ($ 1^{st}, 2^{nd} $ column), in quasi-geostrophic ($ 3^{rd}, 4^{th} $ column), and in incompressible ($ 5^{th}, 6^{th} $ column) regime for regular ($ 1^{st}, 3^{rd}, 5^{th} $ column) and irregular ($ 2^{nd}, 4^{th}, 6^{th} $ column) meshes.
Figure 9.  Initial fluid depth $ D $ and relative potential vorticity $ q_{\rm rel} $ in geostrophic balance. Contours for $ D $ between $ 9.93 {\rm km} $ and $ 10 {\rm km} $ with interval of $ 0.005 {\rm km} $ and for $ q_{\rm rel} $ between $ -0.45 {\rm days^{-1}km^{-1}} $ and $ 1.7 {\rm days^{-1}km^{-1}} $ with interval of $ 0.1 {\rm days^{-1}km^{-1}} $.
Figure 10.  Snapshots of relative potential vorticity $ q_{\rm rel} $ for $ H_0 = 10 $km on regular (upper row) and irregular (lower row) meshes with $ 2 \cdot 256^2 $ cells. Contours between $ -0.45 {\rm days^{-1}km^{-1}} $ and $ 1.7 {\rm days^{-1}km^{-1}} $ with interval of $ 0.1 {\rm days^{-1}km^{-1}} $.
Figure 12.  Vortex interaction test case: relative errors of $ E(t) $ (upper row), of $ PV(t) $ (middle row), and $ PE(t) $ (lower row) for a fluid in semi-geostrophic ($ 1^{st}, 2^{nd} $ column), in quasi-geostrophic ($ 3^{rd}, 4^{th} $ column), and in incompressible ($ 5^{th}, 6^{th} $ column) regime for regular ($ 1^{st}, 3^{rd}, 5^{th} $ column) and irregular ($ 2^{nd}, 4^{th}, 6^{th} $ column) meshes with $ 2 \cdot 256^2 $ cells.
Figure 11.  Comparison of $ q_{\rm rel} $ and $ D $ for fluids in semi-geostrophic (left), quasi-geostrophic (middle), and incompressible regimes (right) for a regular mesh with $ 2 \cdot 256^2 $ cells. Contours for $ D $ between $ -0.12 {\rm km} +H_0 $ and $ 0.02 {\rm km} + H_0 $ with interval of $ 0.01 {\rm km} $. Contours for $ q_{\rm rel} $; left: between $ -13 {\rm days^{-1}km^{-1}} $ and $ 50 {\rm days^{-1}km^{-1}} $ with interval of $ 3 {\rm days^{-1}km^{-1}} $; middle: between $ -7 {\rm days^{-1}km^{-1}} $ and $ 25 {\rm days^{-1}km^{-1}} $ with interval of $ 2 {\rm days^{-1}km^{-1}} $; right: between $ -0.45 {\rm days^{-1}km^{-1}} $ and $ 1.7 {\rm days^{-1}km^{-1}} $ with interval of $ 0.1 {\rm days^{-1}km^{-1}} $.
Figure 13.  Initial fields of fluid depth (left) and relative potential vorticity (right) in geostrophic balance for the shear flow test case on a regular mesh with $ N = 2 \cdot 256^2 $ cells. Contours for $ q_{\rm rel} $ between $ -11 {\rm days^{-1}km^{-1}} $ and $ 11 {\rm days^{-1}km^{-1}} $ with interval of $ 1 {\rm days^{-1}km^{-1}} $, and for $ D $ between $ -0.06 {\rm km} +H_0 $ and $ 0.04 {\rm km} + H_0 $ with interval of $ 0.002 {\rm km} $.
Figure 14.  Shear flow test case: snapshots of relative potential vorticity $ q_{\rm rel} $ on regular (upper row) and irregular (lower row) mesh with $ 2\cdot 256^2 $ cells. Contours between $ -11 {\rm days^{-1}km^{-1}} $ and $ 11 {\rm days^{-1}km^{-1}} $ with interval of $ 2 {\rm days^{-1}km^{-1}} $.
Figure 15.  Shear flow test case: snapshots of $ D $ on regular (upper row) and irregular (lower row) mesh with $ 2\cdot 256^2 $ cells. Contours between $ -0.06 {\rm km} +H_0 $ and $ 0.04 {\rm km} + H_0 $ with interval of $ 0.004 {\rm km} $.
Figure 16.  Shear flow test case: relative errors of total energy $ E(t) $ (upper row), of mass-weighted potential vorticity $ PV(t) $ (middle row), and potential enstrophy $ PE(t) $ (lower row) for a fluid in quasi-geostrophic regime for regular (left) and irregular (right) meshes with $ 2 \cdot 256^2 $ cells.
Table 1.  Continuous and discrete objects
Continuous diffeomorphism Discrete diffeomorphisms
$ \operatorname{Diff}(M)\ni\varphi $ $ \mathsf{D}( \mathbb{M} )\ni q $
Lie algebra Discrete diffeomorphisms
$ \mathfrak{X} (M)\ni\mathbf{u} $ $ \mathfrak{d} ( \mathbb{M} ) \ni A $
Group action on functions Group action on discrete functions
$ f \mapsto f \circ \varphi $ $ F\mapsto q^{-1} F $
Lie algebra action on functions Lie algebra action on discrete functions
$ f\mapsto \mathbf{d} f \cdot \mathbf{u} $ $ F\mapsto -A F $
Group action on densities Group action on discrete densities
$ \rho \mapsto ( \rho \circ \varphi)J \varphi $ $ D\mapsto \Omega^{-1} q^\mathsf{T}\Omega D $
Lie algebra action on densities Lie algebra action on discrete densities
$ \rho \mapsto \operatorname{div}(\rho\mathbf{u} ) $ $ D \mapsto \Omega^{-1} A^\mathsf{T}\Omega D $
Hamilton's principle Lagrande-d'Alembert principle
$ \delta \int_0^T L_{ \rho _0}( \varphi , \dot{\varphi}) dt = 0, $
for arbitrary variations $ \delta \varphi $
$ \delta \int_0^T L_{ D _0}( q , \dot q ) dt = 0 $, $ \dot q q ^{-1} \in \mathcal{S} \cap \mathcal{R} $,
for variations $ \delta q q ^{-1} \in \mathcal{S} \cap \mathcal{R} $
Eulerian velocity and density Eulerian discrete velocity and discrete density
$ \mathbf{u} = \dot{ \varphi } \circ \varphi^{-1} $, $ \rho = (\rho _0 \circ \varphi ^{-1} ) J \varphi^{-1} $ $ A = \dot{ q} q^{-1} $, $ D = \Omega^{-1} q^{-\mathsf{T}}\Omega D_0 $
Euler-Poincaré principle Euler-Poincaré-d'Alembert principle
$ \delta \int_0^T \ell( \mathbf{u} , \rho ) dt = 0 $, $ \delta \mathbf{u} = \partial _t \boldsymbol{\zeta} + [\boldsymbol{\zeta}, \mathbf{u} ] $, $ \delta \rho = - \operatorname{div}( \rho \boldsymbol{\zeta} ) $ $ \delta \int_0^T \ell( A , D ) dt = 0 $, $ \delta A = \partial_t B+[B, A] $,
$ \delta D = - \Omega ^{-1} B^\mathsf{T} \Omega D $,
$ A, B \in \mathcal{S} \cap \mathcal{R} $
Compressible Euler equations Discrete compressible Euler equations
Form Ⅰ:
$ \partial _t ( \rho ( \mathbf{u} ^\flat + \mathbf{R} ^\flat )) + \mathbf{i} _{\rho \mathbf{u} } \omega + \operatorname{div}( \rho\mathbf{u} ) ( \mathbf{u} ^\flat + \mathbf{R} ^\flat ) \\ = - \rho \mathbf{d} \big( \frac{1}{2} | \mathbf{u} | ^2 + \frac{\partial \varepsilon }{\partial \rho } \big) $
Form Ⅰ:
on 2D simplicial gridEquation (43)
Form Ⅱ :
$ \rho\partial _t \mathbf{u} ^\flat + \mathbf{i} _{\rho \mathbf{u} } \omega = - \rho \mathbf{d} \big( \frac{1}{2} | \mathbf{u} | ^2 + \frac{\partial \varepsilon }{\partial \rho } \big) $
Form Ⅱ:
on 2D simplicial gridEquation (40)
Continuous diffeomorphism Discrete diffeomorphisms
$ \operatorname{Diff}(M)\ni\varphi $ $ \mathsf{D}( \mathbb{M} )\ni q $
Lie algebra Discrete diffeomorphisms
$ \mathfrak{X} (M)\ni\mathbf{u} $ $ \mathfrak{d} ( \mathbb{M} ) \ni A $
Group action on functions Group action on discrete functions
$ f \mapsto f \circ \varphi $ $ F\mapsto q^{-1} F $
Lie algebra action on functions Lie algebra action on discrete functions
$ f\mapsto \mathbf{d} f \cdot \mathbf{u} $ $ F\mapsto -A F $
Group action on densities Group action on discrete densities
$ \rho \mapsto ( \rho \circ \varphi)J \varphi $ $ D\mapsto \Omega^{-1} q^\mathsf{T}\Omega D $
Lie algebra action on densities Lie algebra action on discrete densities
$ \rho \mapsto \operatorname{div}(\rho\mathbf{u} ) $ $ D \mapsto \Omega^{-1} A^\mathsf{T}\Omega D $
Hamilton's principle Lagrande-d'Alembert principle
$ \delta \int_0^T L_{ \rho _0}( \varphi , \dot{\varphi}) dt = 0, $
for arbitrary variations $ \delta \varphi $
$ \delta \int_0^T L_{ D _0}( q , \dot q ) dt = 0 $, $ \dot q q ^{-1} \in \mathcal{S} \cap \mathcal{R} $,
for variations $ \delta q q ^{-1} \in \mathcal{S} \cap \mathcal{R} $
Eulerian velocity and density Eulerian discrete velocity and discrete density
$ \mathbf{u} = \dot{ \varphi } \circ \varphi^{-1} $, $ \rho = (\rho _0 \circ \varphi ^{-1} ) J \varphi^{-1} $ $ A = \dot{ q} q^{-1} $, $ D = \Omega^{-1} q^{-\mathsf{T}}\Omega D_0 $
Euler-Poincaré principle Euler-Poincaré-d'Alembert principle
$ \delta \int_0^T \ell( \mathbf{u} , \rho ) dt = 0 $, $ \delta \mathbf{u} = \partial _t \boldsymbol{\zeta} + [\boldsymbol{\zeta}, \mathbf{u} ] $, $ \delta \rho = - \operatorname{div}( \rho \boldsymbol{\zeta} ) $ $ \delta \int_0^T \ell( A , D ) dt = 0 $, $ \delta A = \partial_t B+[B, A] $,
$ \delta D = - \Omega ^{-1} B^\mathsf{T} \Omega D $,
$ A, B \in \mathcal{S} \cap \mathcal{R} $
Compressible Euler equations Discrete compressible Euler equations
Form Ⅰ:
$ \partial _t ( \rho ( \mathbf{u} ^\flat + \mathbf{R} ^\flat )) + \mathbf{i} _{\rho \mathbf{u} } \omega + \operatorname{div}( \rho\mathbf{u} ) ( \mathbf{u} ^\flat + \mathbf{R} ^\flat ) \\ = - \rho \mathbf{d} \big( \frac{1}{2} | \mathbf{u} | ^2 + \frac{\partial \varepsilon }{\partial \rho } \big) $
Form Ⅰ:
on 2D simplicial gridEquation (43)
Form Ⅱ :
$ \rho\partial _t \mathbf{u} ^\flat + \mathbf{i} _{\rho \mathbf{u} } \omega = - \rho \mathbf{d} \big( \frac{1}{2} | \mathbf{u} | ^2 + \frac{\partial \varepsilon }{\partial \rho } \big) $
Form Ⅱ:
on 2D simplicial gridEquation (40)
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