Finite element exterior calculus framework for geophysical fluid dynamics

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1 Finite element exterior calculus framework for geophysical fluid dynamics Colin Cotter Department of Aeronautics Imperial College London Part of ongoing work on UK Gung-Ho Dynamical Core Project funded by NERC/STFC/UK Met Office Colin Cotter

2 Aim Aim of Grids workpackage in GungHo Identify numerical discretisation in horizontal that can be used on pseudo-uniform grids on the sphere, but preserves properties of C-grid finite difference method on latitude-longitude grid. Colin Cotter

3 Aim Aim of talk Identify numerical discretisation in horizontal that are stable and accurate on adaptively refined meshes, but preserves properties of C-grid finite difference method on latitude-longitude grid. Colin Cotter

4 C grids and TRiSK J. Thuburn, Numerical wave propagation on the hexagonal C-grid, JCP J. Thuburn, T. D. Ringler, W. C. Skamarock and J. B. Klemp, Numerical representation of geostrophic modes on arbitrarily structured C-grids, JCP T. D. Ringler, J. Thuburn, J. B. Klemp and W. C. Skamarock, A unified approach to energy conservation and potential vorticity dynamics on arbitrarily-structured C-grids, JCP Colin Cotter

5 Beyond TRiSK J. Thuburn and C. J. Cotter, A framework for mimetic discretization of the rotating shallow-water equations on arbitrary polygonal grids, SIAM J. Sci. Comput Requirements for extension to non-orthogonal grids. C. J. Cotter and J. Shipton, Mixed finite elements for numerical weather prediction, JCP, Mimetic finite element methods applied to linear RSWE are energy conserving, have no spurious pressure modes, and geostrophic modes are steady on f -plane, and allow flexibility to increase order of accuracy, use non-orthogonal grids, tinker with balance between velocity and pressure degrees of freedom to remove spurious mode branches. Colin Cotter

6 Objectives Target features of finite element formulation applied to shallow-water equations on sphere: 1 Overall scheme high(er)-order. 2 Bounded, high-order, local conservation of mass. 3 Stable accurate advection of PV, that is consistent with mass transport. 4 Steady geostrophic linear modes on f-plane. 5 Linear system is energy conserving. Colin Cotter

7 Mimetic properties Extend TRiSK framework to finite element approach. Features we need to import: 1 f = 0 identity preserved (d 2 = 0). 2 f = 0 identity preserved (d 2 = 0). 3 Coriolis term maps f to f (Hodge star). For mimetic finite differences, feature (1) is obtained on the primal grid, and feature (2) is obtained on the dual grid. Feature (3) involves mapping from primal to dual grid. Key difference There is no dual grid (in general) in the finite element approach. We circumvent this by building finite element functions that satisfy (1), obtain (2) by integration by parts, and use finite element projection for (3). 1 1 Watch this space for dual grid FEM schemes! Colin Cotter

8 Mimetic properties Extend TRiSK framework to finite element approach. Features we need to import: 1 f = 0 identity preserved (d 2 = 0). 2 f = 0 identity preserved (d 2 = 0). 3 Coriolis term maps f to f (Hodge star). For mimetic finite differences, feature (1) is obtained on the primal grid, and feature (2) is obtained on the dual grid. Feature (3) involves mapping from primal to dual grid. Key difference There is no dual grid (in general) in the finite element approach. We circumvent this by building finite element functions that satisfy (1), obtain (2) by integration by parts, and use finite element projection for (3). 1 1 Watch this space for dual grid FEM schemes! Colin Cotter

9 Mimetic properties Extend TRiSK framework to finite element approach. Features we need to import: 1 f = 0 identity preserved (d 2 = 0). 2 f = 0 identity preserved (d 2 = 0). 3 Coriolis term maps f to f (Hodge star). For mimetic finite differences, feature (1) is obtained on the primal grid, and feature (2) is obtained on the dual grid. Feature (3) involves mapping from primal to dual grid. Key difference There is no dual grid (in general) in the finite element approach. We circumvent this by building finite element functions that satisfy (1), obtain (2) by integration by parts, and use finite element projection for (3). 1 1 Watch this space for dual grid FEM schemes! Colin Cotter

10 Mimetic properties Extend TRiSK framework to finite element approach. Features we need to import: 1 f = 0 identity preserved (d 2 = 0). 2 f = 0 identity preserved (d 2 = 0). 3 Coriolis term maps f to f (Hodge star). For mimetic finite differences, feature (1) is obtained on the primal grid, and feature (2) is obtained on the dual grid. Feature (3) involves mapping from primal to dual grid. Key difference There is no dual grid (in general) in the finite element approach. We circumvent this by building finite element functions that satisfy (1), obtain (2) by integration by parts, and use finite element projection for (3). 1 1 Watch this space for dual grid FEM schemes! Colin Cotter

11 Mimetic properties Extend TRiSK framework to finite element approach. Features we need to import: 1 f = 0 identity preserved (d 2 = 0). 2 f = 0 identity preserved (d 2 = 0). 3 Coriolis term maps f to f (Hodge star). For mimetic finite differences, feature (1) is obtained on the primal grid, and feature (2) is obtained on the dual grid. Feature (3) involves mapping from primal to dual grid. Key difference There is no dual grid (in general) in the finite element approach. We circumvent this by building finite element functions that satisfy (1), obtain (2) by integration by parts, and use finite element projection for (3). 1 1 Watch this space for dual grid FEM schemes! Colin Cotter

12 Finite element exterior calculus Property (1) built in: E }{{} Continuous S }{{} Continuous normals V }{{} Discontinuous Lose continuity at element boundaries after taking derivatives. See: Arnold, Falk and Winther, Acta Numerica, 2006 for a review. Colin Cotter

13 Some candidate FE spaces E = P2 }{{} Continuous S = BDM1 }{{} Continuous normals V = P0 }{{} Discontinuous/finite volume Same DOF counts as TRiSK on hexagons. Extra branches of Rossby modes (requires analysis). Colin Cotter

14 Some candidate FE spaces E = P2+ }{{} Continuous S = BDFM1 }{{} Continuous normals V = P1 DG }{{} Discontinuous Equal balance of vorticity and pressure DOFs. Second-order accurate, can use DG advection. Colin Cotter

15 Finite element functions Property (1) built in: E }{{} Continuous S }{{} Continuous normals V }{{} Discontinuous Lose continuity at element boundaries after taking derivatives. Property (2) defined weakly (integrate by parts): Weak gradient of D V: Ω w δd dv = Ω wd ds, w S. Weak curl of u S: Ω γδ u ds = Ω γ u ds, γ E. V }{{} Discontinuous δ S }{{} Continuous normals δ E }{{} Continuous Colin Cotter

16 Finite element functions Property (2) defined weakly (integrate by parts): So: Weak gradient of D V: Ω w δd ds = Ω wd ds, w S. Weak curl of u S: Ω γδ u ds = Ω γ u ds, γ E. Ω γδ δd ds = γ δd ds Ω = γ D ds = 0, γ E, Ω }{{} =0 and so δ δd = 0. Colin Cotter

17 PV conservation RSWE in Strong Form: u t + Q }{{} =(ζ+f )u + gd u 2 = 0, }{{} =K D t + (ud) = 0. Finite element discretisation of linear RSWE: Seek u S, D V with w u ds + w Q ds w(gd + K ) ds = 0, t Ω Ω Ω φd ds φ ud ds + φ t Du nds = 0. e for all suitable test functions w S, φ V. e e Colin Cotter

18 Vorticity equation Define vorticity on E using integration by parts: γζ ds = γ u ds, γ E. Ω Ω Get vorticity equation by choosing test function w = γ in velocity equation: d γζ ds = d γ u ds, dt Ω dt Ω = γ Q ds ( γ) (gd + K ) ds Ω Ω }{{} =0 = γ Q ds, γ E. Ω Integration by parts allowed since γ continuous and if Q has continuous normal components. How to construct Q? Colin Cotter

19 How to construct Q? Follow steps of TRiSK: 1 Update D and diagnose a mass flux F. 2 Write PV equation as 3 Set Q = (F q). (qd) + (F q) = 0. t Colin Cotter

20 Compute mass flux DG/finite volume advection equation for D V, φd ds φ ud ds + φ t Du nds = 0, φ V, e e e can be written as for flux F S. t D + F = 0, Crucially: flux can be reconstructed locally (no global elliptic solve), and still works if apply a slope limiter to D to preserve monoticity. (Makes use of Fortin projection). Colin Cotter

21 Compute PV flux Define potential vorticity q E: γqd ds = γ u ds + Ω PV equation is then d γqd ds + dt Ω Ω Ω Ω γf ds, γ E. γ (F q) ds = 0, γ E. For constant q = q 0 we get γ Ω t qd ds = γq 0 D + F Ω } t ds = 0, γ E, {{} =0 so q t = 0 and q stays constant. Colin Cotter

22 Compute PV flux II PV equation (after integration by parts) is d γqd ds γ (F q) ds, γ E. dt Ω Ω Substitute definition of PV, and rewrite as d ( ) ( ) γ u dv + γ (F q) dv. dt Ω Compare with velocity equation with w = γ: d ( ) ( ) γ u dv + γ Q dv, dt so Q = F q. Ω Ω Ω Colin Cotter

23 Stabilising PV advection I What about PV advection? q is in E, not V. Requirement for PV Don t need rigorous bounds for q, just need advection to be stable and not too oscillatory near to fronts, to keep control over the divergence-free component of u. Can add extra dissipative fluxes to PV equation, modifying Q: d γqd ds γ (F q + Q ) ds = 0, γ E, dt Ω with Q = F q + Q. Ω Colin Cotter

24 Stabilising PV advection II Simple Laplacian diffusive fluxes: d γqd ds γ (F q κ q) ds = 0, γ E, dt Ω Ω Q = F q κ q. Petrov-Galerkin method (replaces γ γ + τu γ): d (γ + τu γ) qd ds γ (F q) ds dt Ω Ω + τu γ (F q) ds = 0, γ E, Q = F q τu ( t (qd) + (F q)). Linear, stable, but remains high-order (so can still get overshoots near discontinuities). Ω Colin Cotter

25 Stabilising PV advection II Simple Laplacian diffusive fluxes: d γqd ds γ (F q κ q) ds = 0, γ E, dt Ω Ω Q = F q κ q. Petrov-Galerkin method (replaces γ γ + τu γ): d (γ + τu γ) qd ds γ (F q) ds dt Ω Ω + τu γ (F q) ds = 0, γ E, Q = F q τu ( t (qd) + (F q)). Linear, stable, but remains high-order (so can still get overshoots near discontinuities). Ω Colin Cotter

26 Stabilising PV advection II Simple Laplacian diffusive fluxes: d γqd ds γ (F q κ q) ds = 0, γ E, dt Ω Ω Q = F q κ q. Petrov-Galerkin method (replaces γ γ + τu γ): d (γ + τu γ) qd ds γ (F q) ds dt Ω Ω + τu γ (F q) ds = 0, γ E, Q = F q τu ( t (qd) + (F q)). Linear, stable, but remains high-order (so can still get overshoots near discontinuities). Ω Colin Cotter

27 Stabilising PV advection III If we really need it: Petrov-Galerkin method with discontinuity capturing: d (γ + τu γ) qd ds γ (F q) ds dt Ω Ω + τu γ (F q) ds Ω + η γd q ds = 0, γ E, Where η is a discontinuity detecting switch. Q = F q τu ( t (qd) + (F q)) ηd q. Ω Colin Cotter

28 Timestepping framework Incremental pseudo-newton formulation. Helmholtz equation includes Coriolis term (hybridisation). Mass transport uses subcycled explicit steps with Kuzmin s vertex-based slope limiter. PV advection takes one implicit timestep (to compute Q) with Petrov-Galerkin stabilisation. Colin Cotter

29 Testcases Solid rotation testcase. Colin Cotter

30 Testcases Solid rotation testcase. Colin Cotter

81 Summary and Outlook Summary: TRiSK s mimetic scheme adapted for finite element exterior calculus framework. Based on sequence of spaces for streamfunction, velocity, pressure. Outlook: Stable consistent PV advection built in. Further test cases comparisons with GungHo team. Primal-Dual grid version (see talks by JT). Extension to 3D PV conservation. Colin Cotter

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