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

Previous Article
JCD Home
This Issue
Next Article
Mori-Zwanzig reduced models for uncertainty quantification

June 2019, 6(1): 1-37. doi: 10.3934/jcd.2019001

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

Werner Bauer ^1,2, and François Gay-Balmaz ^3,

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

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.

Keywords: Geometric discretization, structure-preserving schemes, fluid dynamics, Euler-Poincaré formulation, rotating shallow water equations.

Mathematics Subject Classification: Primary: 65P10, 76M60, 37N10; Secondary: 37K05, 37K65.

Citation: Werner Bauer, François Gay-Balmaz. Towards a geometric variational discretization of compressible fluids: The rotating shallow water equations. Journal of Computational Dynamics, 2019, 6 (1) : 1-37. 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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[4]	W. Bauer, M. Baumann, L. Scheck, A. Gassmann, V. 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. Google Scholar
[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. Google Scholar
[6]	W. Bauer and F. Gay-Balmaz, Variational integrators for the anelastic and pseudo-incompressible flows, preprint, 2017, arXiv: 1701.06448.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[11]	M. J. P. Cullen, A Mathematical Theory of Large-scale Atmosphere/ocean Flow, Imperial College Press, London, 2006. doi: 10.1142/p375. Google Scholar
[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. Google Scholar
[13]	F. Demoures, F. Gay-Balmaz, M. 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. Google Scholar
[14]	M. Desbrun, E. Gawlik, F. 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. Google Scholar
[15]	A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, 2004. doi: 10.1007/978-1-4757-4355-5. Google Scholar
[16]	E. Gawlik, P. Mullen, D. Pavlov, J. 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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	D. D. Holm, J. 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. Google Scholar
[21]	A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators, Arch. Rat. Mech. Anal., 167 (2003), 85-146. doi: 10.1007/s00205-002-0212-y. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X. Google Scholar
[26]	J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505. Google Scholar
[27]	D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. 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. Google Scholar
[28]	J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, New York, 1979.Google Scholar
[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. Google Scholar
[30]	S. Reich, N. Wood and A. Staniforth, Semi-implicit methods, nonlinear balance, and regularized equations, Atmospheric Science Letters, 8 (2007), 1-6. doi: 10.1002/asl.142. Google Scholar
[31]	T. D. Ringler, J. Thuburn, J. 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. Google Scholar
[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. Google Scholar
[33]	A. Staniforth, N. 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. Google Scholar
[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. Google Scholar
[35]	J. Thuburn, T. D. Ringler, W. 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. Google Scholar
[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. Google Scholar
[37]	D. L. Williamson, J. B. Drake, J. J. Hack, R. 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. Google Scholar
[38]	V. Zeitlin (Ed.), Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances, Elsevier, New York, 2007.Google Scholar

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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[4]	W. Bauer, M. Baumann, L. Scheck, A. Gassmann, V. 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. Google Scholar
[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. Google Scholar
[6]	W. Bauer and F. Gay-Balmaz, Variational integrators for the anelastic and pseudo-incompressible flows, preprint, 2017, arXiv: 1701.06448.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[11]	M. J. P. Cullen, A Mathematical Theory of Large-scale Atmosphere/ocean Flow, Imperial College Press, London, 2006. doi: 10.1142/p375. Google Scholar
[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. Google Scholar
[13]	F. Demoures, F. Gay-Balmaz, M. 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. Google Scholar
[14]	M. Desbrun, E. Gawlik, F. 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. Google Scholar
[15]	A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, 2004. doi: 10.1007/978-1-4757-4355-5. Google Scholar
[16]	E. Gawlik, P. Mullen, D. Pavlov, J. 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. Google Scholar
[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.Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	D. D. Holm, J. 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. Google Scholar
[21]	A. Lew, J. E. Marsden, M. Ortiz and M. West, Asynchronous variational integrators, Arch. Rat. Mech. Anal., 167 (2003), 85-146. doi: 10.1007/s00205-002-0212-y. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514. doi: 10.1017/S096249290100006X. Google Scholar
[26]	J. E. Marsden, G. W. Patrick and S. Shkoller, Multisymplectic geometry, variational integrators and nonlinear PDEs, Comm. Math. Phys., 199 (1998), 351-395. doi: 10.1007/s002200050505. Google Scholar
[27]	D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. 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. Google Scholar
[28]	J. Pedlosky, Geophysical Fluid Dynamics, Springer Verlag, New York, 1979.Google Scholar
[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. Google Scholar
[30]	S. Reich, N. Wood and A. Staniforth, Semi-implicit methods, nonlinear balance, and regularized equations, Atmospheric Science Letters, 8 (2007), 1-6. doi: 10.1002/asl.142. Google Scholar
[31]	T. D. Ringler, J. Thuburn, J. 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. Google Scholar
[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. Google Scholar
[33]	A. Staniforth, N. 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. Google Scholar
[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. Google Scholar
[35]	J. Thuburn, T. D. Ringler, W. 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. Google Scholar
[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. Google Scholar
[37]	D. L. Williamson, J. B. Drake, J. J. Hack, R. 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. Google Scholar
[38]	V. Zeitlin (Ed.), Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances, Elsevier, New York, 2007.Google Scholar

Figure 1. Notation and indexing conventions for the 2D simplicial mesh.

Download full-size image

Download as PowerPoint slide

Figure 2. Regular mesh with equilateral triangles and irregular mesh with central refinement region, both with $ 2 \cdot 32^2 $ triangular grid cells.

Download full-size image

Download as PowerPoint slide

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.

Download full-size image

Download as PowerPoint slide

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).

Download full-size image

Download as PowerPoint slide

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} $.

Download full-size image

Download as PowerPoint slide

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}} $.

Download full-size image

Download as PowerPoint slide

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.

Download full-size image

Download as PowerPoint slide

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.

Download full-size image

Download as PowerPoint slide

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}} $.

Download full-size image

Download as PowerPoint slide

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}} $.

Download full-size image

Download as PowerPoint slide

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.

Download full-size image

Download as PowerPoint slide

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}} $.

Download full-size image

Download as PowerPoint slide

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} $.

Download full-size image

Download as PowerPoint slide

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}} $.

Download full-size image

Download as PowerPoint slide

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} $.

Download full-size image

Download as PowerPoint slide

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.

Download full-size image

Download as PowerPoint slide

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)

Download as excel

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)

[1]	Emanuel-Ciprian Cismas. Euler-Poincaré-Arnold equations on semi-direct products II. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5993-6022. doi: 10.3934/dcds.2016063
[2]	Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2019146
[3]	Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093
[4]	Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[5]	Jeffrey K. Lawson, Tanya Schmah, Cristina Stoica. Euler-Poincaré reduction for systems with configuration space isotropy. Journal of Geometric Mechanics, 2011, 3 (2) : 261-275. doi: 10.3934/jgm.2011.3.261
[6]	Vikas S. Krishnamurthy. The vorticity equation on a rotating sphere and the shallow fluid approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6261-6276. doi: 10.3934/dcds.2019273
[7]	Marcel Oliver, Sergiy Vasylkevych. Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 827-846. doi: 10.3934/dcds.2011.31.827
[8]	Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[9]	Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
[10]	Anwar Ja'afar Mohamad Jawad, Mohammad Mirzazadeh, Anjan Biswas. Dynamics of shallow water waves with Gardner-Kadomtsev-Petviashvili equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1155-1164. doi: 10.3934/dcdss.2015.8.1155
[11]	Andrei Cozma, Christoph Reisinger. Exponential integrability properties of Euler discretization schemes for the Cox--Ingersoll--Ross process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3359-3377. doi: 10.3934/dcdsb.2016101
[12]	Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473
[13]	Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[14]	Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[15]	Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799
[16]	Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
[17]	Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375
[18]	David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629
[19]	Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331
[20]	Nikolaos Halidias. Construction of positivity preserving numerical schemes for some multidimensional stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 153-160. doi: 10.3934/dcdsb.2015.20.153

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Werner Bauer François Gay-Balmaz
on Google Scholar
Werner Bauer François Gay-Balmaz

Impact Factor:

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Werner Bauer François Gay-Balmaz
on Google Scholar
Werner Bauer François Gay-Balmaz

[Back to Top]