Mixed Mimetic Spectral Elements for Geophysical Fluid Dynamics - TOY... · Mixed Mimetic Spectral...

Mixed Mimetic Spectral Elements for Geophysical FluidDynamics

Dave Lee

Los Alamos National Laboratory

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

Outline

I Connection of finite volumes to differential forms

I Key ideas of differential forms

I Differential forms for discrete data

I Construction of mixed mimetic spectral elements

I Rotating shallow water equations

I Results

I Outlook and future directions


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

Discrete versions of the following hold exactly

I ∇×∇φ = 0

I ∇ · ∇⊥ψ = 0

I∫∇× uda =

∮u · dl

I∫∇ · udv =

∫u · da

Desirable properties for geophysical modelling

I Conservation of moments (mass, vorticity, energy, potential enstrophy...)

I Stationary geostrophic modes: f k× u + 1ρ∇p = 0

I 2/1 ratio of velocity to pressure degrees of freedom (two gravity waves for everyRossby wave)

See Thuburn (2008) JCP Some conservation issues for the dynamical cores of NWPand climate models for a detailed discussion


Finite volumes and differential forms

b

b

ω2ψ0i

~u1

~u1

p2φ0,i

ψ0 → ∇⊥ → ~u1 → ∇· → p2

ω2 ← ∇× ← ~u1 ← ∇← φ0

l ⋆ l ⋆ l ⋆

b

bψ0i+1

φ0,i+1


Key concepts of differential forms

Exterior derivative d, generalization of ∇,∇·,∇×,∇⊥I Maps from a k to a k + 1 space Λk → Λk+1

I d d = 0, R→ Λ0 → Λ1 → Λ2 = 0 (∇×∇φ = 0, ∇ · ∇⊥ψ = 0)

Boundary operator ∂I Maps to the boundary of a k form Λk → Λk−1

Hodge star operator ?I Maps from a k to an n − k space, Λk → Λn−k

I Analogous to mapping between Voronoi and Delaunay grids on a finite volumemesh

Λ0 → d→ Λ1 → d→ Λ2

Λ2 ← d← Λ1 ← d← Λ0

l ⋆ l ⋆ l ⋆

Generalized Stokes theorem∫Ωk

dαk−1 =

∫∂Ωk

αk−1 → (dαk−1,Ωk ) = (αk−1, ∂Ωk )

I k = 1: Fundamental theorem of calculusI k = 2: Kelvin-Stokes circulation theoremI k = 3: Gauss’ divergence theorem


Application to spectral elements

I The standard (0 form) spectral element basis is given as f 0(ξ) = a0i li (ξ)

I Start from the premise that in 1D we wish to exactly satisfy the fundamental th.of calculus between nodes ξi and ξi+1:∫ ξi+1

ξi

df 0(ξ) = f 0(ξi+1)− f 0(ξi ) = a0i+1 − a0

i = b1i =

∫ ξi+1

ξi

g1(ξ)

I The corresponding 1 form edge function expansion is then given asg1(ξ) = b1

i ei (ξ) subject to

df 0 = g1

I Edge functions are orthogonal with respect to the set of 1 forms such that∫ ξi+1

ξi

ej (ξ) = δi,j

I Satisfies the Kelvin-Stokes and Gauss-divergence theorems in higher dimensionsvia tensor product combinations of li (ξ) and ej (ξ)


Algebraic topology and the discrete exterior derivative

E1,2 =

+1 0 0 00 +1 0 0−1 0 +1 00 −1 0 +10 0 −1 00 0 0 −1−1 0 0 0+1 −1 0 00 +1 0 00 0 −1 00 0 +1 −10 0 0 +1

I Domain is broken up into discrete k forms called chains

I Chains are topological objects with no measure

I Data that resides on these k-chains is referred to as co-chains

I Orientation of k forms with respect to the k − 1 forms given by the discreteboundary operator Ek−1,k

I ∂ ∂ = 0 holds for discrete case: Ek−2,k−1Ek−1,k = 0.

I Discrete exterior derivative is the transpose of the boundary operator:E k,k−1 = ET

k−1,k


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 )u0j = Mi,ju

0j p1(ξi ) = ej (ξi )p

1j = Ni,jp

1j

First equation may be solved in the strong form as

dp1i

dt= −du0

j

= −E1,0i,j u

0j

= −(u0j+1 − u0

j )

Second equation may be solved in the weak form via the adjoint relation

d

dt(la, lb)Ωk

u0b = −(la, d ? ec )Ωk

p1c

= (dla, ec )Ωkp1c + B.C .s

= E0,1,a,d (ed , ec )Ωkp1c + B.C .s


The shallow water equations

∂u1

∂t= −q0 ∧ F1 − d ? (K2 + gh2)

∂h2

∂t= −dF1

?h2 ∧ q0 = d ? u1 + f 0 F1 = ?h2 ∧ u1 K2 =1

2? u1 ∧ u1

I Define the spaces α0i ∈ 0 forms, β1

i ∈ 1 forms (vector fields), γ2i ∈ 2 forms

I Solve for the diagnostic equations

(α0, ?h2 ∧ q0)Ωk= (α0, d ? ~u1)Ωk

+ (α0, f 0)Ωk

= −E0,1(β1, u1)Ωk+ (α0

a, f0)Ωk

(β1,F1)Ωk= (β1, ?h2 ∧ u1)Ωk

(γ2,K2)Ωk=

1

2(γ2, ?u1 ∧ u1)Ωk

I Solve the prognostic equations

d

dt(β1, u1)Ωk

= −(β1, q0 ∧ F1)Ωk− (β1, d ? (K2 + gh2))Ωk

= −(β1, q0 ∧ F1)Ωk+ E1,2(γ2, (K2 + gh2))Ωk

d

dth2 = −E2,1F1


Results: Spectral convergence

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


Results: Conservation properties

I Mass: exact and holds pointwise, since continuity equation is expressed in thestrong form

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

d

dt(h2,K2 + 0.5gh2)Ωk

= 0

I Potential enstrophy: holds globally to truncation error in time, requires that theterms ?h2 ∧ q0 and q0 ∧ F1 are integrated exactly


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)


Computational cost

I Continuity, ddth2 = −E2,1F1:

I Exact and pointwise, h2,n+1i,j = h2,n

i,j − ∆t(F 1,x,ni+1,j − F 1,x,n

i,j + F 1,y,ni,j+1 − F 1,y,n

i,j )I Very fast!!

I Kinetic energy, (γ2,K2)Ωk= 1

2(γ2, ?u1 ∧ u1)Ωk

:

I Function space γ2 is discontinuous at element boundaries - DGI Fast!

I Potential vorticity, (α0a, ?h

2 ∧ q0)Ωk= −E0,1(β1, u1)Ωk

+ (α0a, f

0)Ωk:

I Use inexact integration (and sacrifice potential enstrophy conservation)I α0 is orthogonal at the (inexact) quadrature points, LHS matrix is diagonalI Fast!

I Velocity, u1 and momentum F1:I Function space β1 has continuous normal componentsI Requires global mass matrix solve - CGI Slow!!


Outlook

I Can we get exact conservation of energy independent of time step viasemi-implicit, staggered time stepping as has been done for Navier Stokes?[Palha and Gerritsma, (2017), JCP]

I Should we be prognosing the vorticity rather than diagnosing?

I Can we do this on the sphere via isoparametric mapping from computationaldomain? - Should be ok since volume, vorticity and energy conservation hold forinexact integration.

I Conservation of potential vorticity requires exact integration - can we do this withinexact integration? - do we even care?

I Turbulence closure: Anticipated Potential Vorticity Method, dissipates enstrophybut preserves energy conservation [Sadourny and Basdevant, (1985), JAS].


References

Geostrophically balanced finite volumes:

I Thuburn, Ringler, Skamarock and Klemp (2010) JCP

I Ringler, Thuburn, Klemp and Skamarock (2010) JCP

I Peixoto (2016), JCP

Mimetic finite elements for geophysical flows:

I Cotter and Shipton (2012) JCP

I Cotter and Thuburn (2014) JCP

I McRae and Cotter (2014) QJRMS

I Thuburn and Cotter (2015) JCP

Spectral elements (a-grid):

I Taylor and Fournier (2010) JCP

I Melvin, Staniforth and Thuburn (2012) QJRMS

Mixed spectral elements:

I Palha and Gerritsma (2011) Spectral and High Order Methods for PartialDifferential Equations

I Kreeft and Gerritsma (2013) JCP

I Palha, Rebelo and Gerritsma (2013) Mimetic Spectral Element Advection

I Hiemstra, Toshniwal, Huijsmans and Gerritsma (2014) JCP


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