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

Werner Bauer; François Gay-Balmaz

doi:10.3934/jcd.2019001

Article Contents

2019, Volume 6, Issue 1: 1-37. Doi: 10.3934/jcd.2019001

This issue Previous Article Next Article Mori-Zwanzig reduced models for uncertainty quantification

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: July 2019

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Abstract

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:

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

Citation:

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

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

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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.

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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).

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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} $.

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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}} $.

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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.

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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.

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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}} $.

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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}} $.

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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.

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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}} $.

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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} $.

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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}} $.

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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} $.

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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.

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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)

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