Mixed Mimetic Spectral Elements for Geophysical Fluid Dynamics Dave - - PowerPoint PPT Presentation

Mixed Mimetic Spectral Elements for Geophysical Fluid Dynamics

Dave Lee

Los Alamos National Laboratory Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Outline

◮ Connection of finite volumes to differential forms ◮ Key ideas of differential forms ◮ Differential forms for discrete data ◮ Construction of mixed mimetic spectral elements ◮ Rotating shallow water equations ◮ Results ◮ Outlook and future directions Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

What is a mimetic method...and why do we care?

Discrete versions of the following hold exactly

◮ ∇ × ∇φ = 0 ◮ ∇ · ∇⊥ψ = 0 ◮

∇ × uda =

• u · dl

◮

∇ · udv =

• u · da

Desirable properties for geophysical modelling

◮ Conservation of moments (mass, vorticity, energy, potential enstrophy...) ◮ Stationary geostrophic modes: f ˆ

k × u + 1

ρ∇p = 0

◮ 2/1 ratio of velocity to pressure degrees of freedom (two gravity waves for every

Rossby wave) See Thuburn (2008) JCP Some conservation issues for the dynamical cores of NWP and climate models for a detailed discussion

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Finite volumes and differential forms

b b

ω2 ψ0

i

• u1
• u1

p2 φ0,i

ψ0 → ∇⊥ → u1 → ∇· → p2 ω2 ← ∇× ← u1 ← ∇ ← φ0 ⋆ ⋆ ⋆

b b

ψ0

i+1

φ0,i+1

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Key concepts of differential forms

Exterior derivative d, generalization of ∇, ∇·, ∇×, ∇⊥

◮ Maps from a k to a k + 1 space Λk → Λk+1 ◮ d ◦ d = 0, R → Λ0 → Λ1 → Λ2 = 0 (∇ × ∇φ = 0, ∇ · ∇⊥ψ = 0)

Boundary operator ∂

◮ Maps to the boundary of a k form Λk → Λk−1

Hodge star operator ⋆

◮ Maps from a k to an n − k space, Λk → Λn−k ◮ Analogous to mapping between Voronoi and Delaunay grids on a finite volume

mesh

Λ0 → d → Λ1 → d → Λ2 Λ2 ← d ← Λ1 ← d ← Λ0 ⋆ ⋆ ⋆

Generalized Stokes theorem

• Ωk

dαk−1 =

• ∂Ωk αk−1 → (dαk−1, Ωk) = (αk−1, ∂Ωk)

◮ k = 1: Fundamental theorem of calculus ◮ k = 2: Kelvin-Stokes circulation theorem ◮ k = 3: Gauss’ divergence theorem Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Application to spectral elements

◮ The standard (0 form) spectral element basis is given as f 0(ξ) = a0

i li(ξ)

◮ Start from the premise that in 1D we wish to exactly satisfy the fundamental th.

• f calculus between nodes ξi and ξi+1:

ξi+1

ξi

df 0(ξ) = f 0(ξi+1) − f 0(ξi) = a0

i+1 − a0 i = b1 i =

ξi+1

ξi

g1(ξ)

◮ The corresponding 1 form edge function expansion is then given as

g1(ξ) = b1

i ei(ξ) subject to

df 0 = g1

◮ Edge functions are orthogonal with respect to the set of 1 forms such that

ξi+1

ξi

ej(ξ) = δi,j

◮ Satisfies the Kelvin-Stokes and Gauss-divergence theorems in higher dimensions

via tensor product combinations of li(ξ) and ej(ξ)

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Algebraic topology and the discrete exterior derivative

E1,2 =                   +1 +1 −1 +1 −1 +1 −1 −1 −1 +1 −1 +1 −1 +1 −1 +1                  

◮ Domain is broken up into discrete k forms called chains ◮ Chains are topological objects with no measure ◮ Data that resides on these k-chains is referred to as co-chains ◮ Orientation of k forms with respect to the k − 1 forms given by the discrete

boundary operator Ek−1,k

◮ ∂ ◦ ∂ = 0 holds for discrete case: Ek−2,k−1Ek−1,k = 0. ◮ Discrete exterior derivative is the transpose of the boundary operator:

E k,k−1 = E T

k−1,k

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Example: 1D wave equation

∂p1 ∂t = −du0 ∂u0 ∂t = −d ⋆ p1 Discretize velocity (0-form) and pressure (1-form) within each element u0(ξi) = lj(ξi)u0

j = Mi,ju0 j

p1(ξi) = ej(ξi)p1

j = Ni,jp1 j

First equation may be solved in the strong form as dp1

i

dt = −du0

j

= −E 1,0

i,j u0 j

= −(u0

j+1 − u0 j )

Second equation may be solved in the weak form via the adjoint relation d dt (la, lb)Ωk u0

b = −(la, d ⋆ ec)Ωk p1 c

= (dla, ec)Ωk p1

c + B.C.s

= E0,1,a,d(ed, ec)Ωk p1

c + B.C.s

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

The shallow water equations

∂u1 ∂t = −q0 ∧ F1 − d ⋆ (K 2 + gh2) ∂h2 ∂t = −dF1 ⋆h2 ∧ q0 = d ⋆ u1 + f 0 F1 = ⋆h2 ∧ u1 K 2 = 1 2 ⋆ u1 ∧ u1

◮ Define the spaces α0

i ∈ 0 forms, β1 i ∈ 1 forms (vector fields), γ2 i ∈ 2 forms

◮ Solve for the diagnostic equations

(α0, ⋆h2 ∧ q0)Ωk = (α0, d ⋆ u1)Ωk + (α0, f 0)Ωk = −E0,1(β1, u1)Ωk + (α0

a, f 0)Ωk

(β1, F1)Ωk = (β1, ⋆h2 ∧ u1)Ωk (γ2, K 2)Ωk = 1 2 (γ2, ⋆u1 ∧ u1)Ωk

◮ Solve the prognostic equations

d dt (β1, u1)Ωk = −(β1, q0 ∧ F1)Ωk − (β1, d ⋆ (K 2 + gh2))Ωk = −(β1, q0 ∧ F1)Ωk + E1,2(γ2, (K 2 + gh2))Ωk d dt h2 = −E 2,1F1

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Results: Spectral convergence

Diagnostic equations converge at their spectral order of accuracy for 3rd (left) and 4th (right) order basis functions

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Results: Conservation properties

◮ Mass: exact and holds pointwise, since continuity equation is expressed in the

strong form

◮ Vorticity: holds globally in the weak form, since E0,1E1,2 = 0 (∇ × ∇ = 0) ◮ Energy: holds globally to truncation error in time for

d dt (h2, K 2 + 0.5gh2)Ωk = 0

◮ Potential enstrophy: holds globally to truncation error in time, requires that the

terms ⋆h2 ∧ q0 and q0 ∧ F1 are integrated exactly

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Results: Vortices from geostrophic balance

ψ = e−2.5((x−2π/3)2+(y−π)2) + e−2.5((x−4π/3)2+(y−π))2

• u = k × ∇ψ

f × u + g∇h = 0 ⇒ h = (f /g)ψ + H f = g = H = 8(>> 1)

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Computational cost

◮ Continuity,

d dt h2 = −E 2,1F1:

◮ Exact and pointwise, h2,n+1 i,j

= h2,n

i,j − ∆t(F 1,x,n i+1,j − F 1,x,n i,j

+ F 1,y,n

i,j+1 − F 1,y,n i,j

)

◮ Very fast!! ◮ Kinetic energy, (γ2, K 2)Ωk = 1

2 (γ2, ⋆u1 ∧ u1)Ωk :

◮ Function space γ2 is discontinuous at element boundaries - DG ◮ Fast! ◮ Potential vorticity, (α0

a, ⋆h2 ∧ q0)Ωk = −E0,1(β1, u1)Ωk + (α0 a, f 0)Ωk :

◮ Use inexact integration (and sacrifice potential enstrophy conservation) ◮ α0 is orthogonal at the (inexact) quadrature points, LHS matrix is diagonal ◮ Fast! ◮ Velocity, u1 and momentum F1: ◮ Function space β1 has continuous normal components ◮ Requires global mass matrix solve - CG ◮ Slow!! Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

Outlook

◮ Can we get exact conservation of energy independent of time step via

semi-implicit, staggered time stepping as has been done for Navier Stokes? [Palha and Gerritsma, (2017), JCP]

◮ Should we be prognosing the vorticity rather than diagnosing? ◮ Can we do this on the sphere via isoparametric mapping from computational

domain? - Should be ok since volume, vorticity and energy conservation hold for inexact integration.

◮ Conservation of potential vorticity requires exact integration - can we do this with

inexact integration? - do we even care?

◮ Turbulence closure: Anticipated Potential Vorticity Method, dissipates enstrophy

but preserves energy conservation [Sadourny and Basdevant, (1985), JAS].

Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements

References

Geostrophically balanced finite volumes:

◮ Thuburn, Ringler, Skamarock and Klemp (2010) JCP ◮ Ringler, Thuburn, Klemp and Skamarock (2010) JCP ◮ Peixoto (2016), JCP

Mimetic finite elements for geophysical flows:

◮ Cotter and Shipton (2012) JCP ◮ Cotter and Thuburn (2014) JCP ◮ McRae and Cotter (2014) QJRMS ◮ Thuburn and Cotter (2015) JCP

Spectral elements (a-grid):

◮ Taylor and Fournier (2010) JCP ◮ Melvin, Staniforth and Thuburn (2012) QJRMS

Mixed spectral elements:

◮ Palha and Gerritsma (2011) Spectral and High Order Methods for Partial

Differential Equations

◮ Kreeft and Gerritsma (2013) JCP ◮ Palha, Rebelo and Gerritsma (2013) Mimetic Spectral Element Advection ◮ Hiemstra, Toshniwal, Huijsmans and Gerritsma (2014) JCP Lee; LA-UR-17-24044 Mixed Mimetic Spectral Elements