Mixed Mimetic Spectral Elements for Atmospheric Simulation Dave Lee - PowerPoint PPT Presentation

Mixed Mimetic Spectral Elements for Atmospheric Simulation Dave Lee (Monash University) February 11, 2020 Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Motivation Improved representation of dynamical processes and long term statistics ◮ Preservation of leading order balance relations ◮ Geostrophic balance (horizontal), f ˆ z × u = −∇ h ◮ Hydrostatic balance (vertical), θ ∂ Π ∂ z = − g ◮ Preservation of conservation laws ◮ Mass ◮ Vorticity ◮ Energy (balanced exchanges) ◮ Potential enstrophy (for exact integration only) Structure preserving formulations � � � Solving PDEs (for geophysical flows) � � � Mimetic discretisations Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

b b b b Mimetic discretisations, a topological perspective φ 0 ,i +1 � u 1 ψ 0 ψ 0 ω 2 i +1 i u 1 � ψ 0 → ∇ ⊥ → � u 1 → ∇· → p 2 � ⋆ � ⋆ � ⋆ ω 2 ← ∇× ← � u 1 ← ∇ ← φ 0 p 2 φ 0 ,i ∇ × ∇ φ = 0 dd u = 0 ∇ · ∇ ⊥ ψ = 0 � � ∇ × u d a = u · d l � ∂ Ω u = � Ω d u � � ∇ · u dv = u · d a Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Mixed mimetic spectral elements (1D) ◮ The standard (nodal) spectral element basis is given as f 0 ( ξ ) = a 0 i l i ( ξ ) ◮ Introduce a secondary edge basis [Gerritsma, 2011] as g 1 ( ξ ) = b 1 i e i ( ξ ) ◮ Integration of the edge functions between GLL nodes are orthogonal such that � ξ i +1 e j ( ξ ) = δ i , j ξ i ◮ Start from the premise that in 1D we wish to exactly satisfy the fundamental theorem of calculus between nodes ξ i and ξ i +1 : � ξ i +1 � ξ i +1 df 0 ( ξ ) = f 0 ( ξ i +1 ) − f 0 ( ξ i ) = a 0 i +1 − a 0 i = b 1 g 1 ( ξ ) i = d ξ ξ i ξ i 3 3 2.5 2.5 2 2 1.5 1.5 1 1 l i ( ξ ) e i ( ξ ) 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 ξ ξ Figure: Nodal (left) and edge (right) functions for a 4 th order spectral element Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Mixed mimetic spectral elements (2D) Via tensor product combinations of l i ( ξ ) and e i ( ξ ) we define the function spaces: i , j ( ξ ) = l i ( ξ ) ⊗ l j ( η ) ∈ V 0 ; C 0 continuous across elements ◮ α 0 i , j ( ξ ) = { l i ( ξ ) ⊗ e j ( η ) , e i ( ξ ) ⊗ l j ( η ) } ∈ V 1 ; C 0 normal components ◮ β 1 i . j ( ξ ) = e i ( ξ ) ⊗ e j ( η ) ∈ V 2 discontinuous across elements ◮ γ 2 And basis function expansions: ◮ ψ 0 i , j = ˆ ψ i , j α 0 i , j u i , j β 1 ,ξ v i , j β 1 ,η ◮ u 1 i , j = { ˆ i , j , ˆ i , j } ◮ p 2 p i , j γ 1 i , j = ˆ i , j In the strong form: u 1 = ∇ ⊥ ψ 0 = h = { ( ˆ ψ i , j − 1 − ˆ ψ i , j ) β 1 ,ξ i , j , ( ˆ ψ i , j − ˆ ψ i − 1 , j ) β 1 ,η h E 1 , 0 ˆ ⇒ u 1 i , j } = β 1 ψ 0 h p 2 = ∇ · u 1 = ⇒ p 2 v i , j − 1 ) γ 0 i , j = γ 2 h E 2 , 1 ˆ u 1 h = (ˆ u i , j − ˆ u i − 1 , j + ˆ v i , j − ˆ h Weak form adjoint relations: � α 0 , ∇ × u 1 � � ∇ ⊥ α 0 , u 1 � � � k , i ) ⊤ � � j = − (E 1 , 0 α 0 i , α 0 ω 0 β 1 k , β 1 u 1 Ω = − Ω = ⇒ ˆ ˆ j l l Ω a Ω a � β 1 , ∇ φ 2 � � ∇ · β 1 , φ 2 � � � k , i ) ⊤ � � j = − (E 2 , 1 ˆ β 1 i , β 1 u 1 γ 2 k , γ 2 φ 2 Ω = − Ω = ⇒ ˆ j l l Ω a Ω a Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

b b b b 2D Hydrodynamics in H ( div ) φ 0 ,i +1 � u 1 ψ 0 ψ 0 ω 2 i +1 i u 1 � ψ 0 → ∇ ⊥ → � u 1 → ∇· → p 2 � ⋆ � ⋆ � ⋆ ω 2 ← ∇× ← � u 1 ← ∇ ← φ 0 φ 0 ,i p 2 Strong form (pointwise) Weak form (Galerkin proj.) ∇ ⊥ , E 1 , 0 ∇× , − (E 1 , 0 i , j : V 0 V 1 j , i ) ⊤ : V 1 V 0 → → � α 0 , ∇ × u 1 � � ∇ ⊥ α 0 , u 1 � u 1 ∇ ⊥ ψ 0 = Ω = − Ω � � k , i ) ⊤ � � E 1 , 0 j = − (E 1 , 0 u 1 i , j ψ 0 α 0 i , α 0 ω 0 β 1 k , β 1 u 1 = ˆ ˆ l i j j l Ω a Ω a ∇· , E 2 , 1 : V 1 V 2 ∇ , − (E 2 , 1 ) ⊤ : V 2 V 1 → → � β 1 , ∇ φ 2 � � ∇ · β 1 , u 1 � p 2 ∇ · u 1 = Ω = − Ω � � k , i ) ⊤ � � E 2 , 1 = − (E 2 , 1 ˆ p 2 i , j u 1 β 1 i , β 1 u 1 γ 2 k , γ 2 φ 2 = ˆ i j j j l l Ω a Ω a 0, E 2 , 1 i , j E 1 , 0 ∇ × ∇ := 0, (E 1 , 0 j , i ) ⊤ (E 2 , 1 k , i ) ⊤ = 0 ∇ · ∇ ⊥ := = 0 j , k Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Structure preserving formulations and conservation laws ◮ Let H be an invariant of the PDE with dependent variables a ◮ Write the PDE as ∂ a ∂ t = S δ H δ a where S is a skew-symmetric operator ◮ H is conserved as ∂ H ∂ t = δ H δ a · ∂ a ∂ t = δ H δ a · S δ H δ a = 0 ◮ Corresponds to the anti-symmetry of the Poisson bracket � A , S B � = { A , B } = −{ B , A } ⇒ {H , H} = −{H , H} = 0 Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Example: Rotating Shallow Water Equations (continuous form) � 1 2 h u · u + g 2 h 2 d Ω H = δ H δ u = h u = F δ H δ h = 1 2 u · u + gh = Φ ∂ � u � � ( ω + f ) / h × � � F � ∇ = − h ∇· 0 Φ ∂ t ∂ H ∂ t = [ F , Φ] · ∂ � u � � ( ω + f ) / h × � � F � ∇ = − [ F , Φ] = 0 h ∇· 0 Φ ∂ t Semi-discrete: � u n +1 − u n � � ¯ 1 � � ( ω + f ) / h × ∇ � F = − h n +1 − h n ¯ ∇· 0 Φ ∆ t F = 1 3 h n u n + 1 6 h n u n +1 + 1 6 h n +1 u n + 1 ¯ 3 h n +1 u n +1 Φ = 1 6 u n · u n + 1 6 u n · u n +1 + 1 6 u n +1 · u n +1 + g 2 h n + g ¯ 2 h n +1 Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Example: Rotating Shallow Water Equations (discrete form) Discrete variational derivatives may be computed as H h ( a h + ǫ b h ) − H h ( a h ) b h , δ H h � � ∀ b h ∈ V 0 / V 1 / V 2 lim = ǫ δ a h ǫ → 0 Such that β h , δ H h � � = � β h , β h � ˆ ∀ β h ∈ V 1 F h = � β h , h h u h � δ u h γ h , δ H h Φ h = � γ h , 1 � � = � γ h , γ h � ˆ ∀ γ h ∈ V 2 2 u h · u h + gh h � δ h h We also define the potential vorticity, q h as: � α h , h h q h � = − (E 1 , 0 ) ⊤ � β h , u h � + � α h , f h � ∀ α h ∈ V 0 � � ˆ � � β h , u n +1 1 − u n � � − (E 2 , 1 ) ⊤ � γ h , γ h � � h � � β h , q h × β h � F h h = − � γ h , h n +1 � γ h , γ h � E 2 , 1 ˆ − h n h � 0 Φ h ∆ t h Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Rotating Shallow Water Equations: Other Conservation Properties ◮ Conservation of mass, ˆ h ˆ 1 ⊤ h h : holds pointwise due to strong form divergence h E 2 , 1 ˆ ˆ 1 ⊤ F h = 0 ◮ Conservation of vorticity, ˆ 1 ⊤ h � α h , α h � ˆ ω h : holds in the weak form as ˆ 1 ⊤ h (E 1 , 0 ) ⊤ � β h , q h × β h � ˆ F h = 0 (E 1 , 0 ) ⊤ (E 2 , 1 ) ⊤ = 0 ω h = − (E 1 , 0 ) ⊤ � β h , β h � ˆ � α h , α h � ˆ u h ◮ Conservation of potential enstrophy, ˆ q h � α h , h h α h � ˆ q h : holds in the weak form for exact integration only : The product rule q × ∇ ⊥ q � = 1 2 ∇ q 2 holds only for exact integration Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

Rotating Shallow Water Equations (results) Williamson TC2, conservation error, vorticity (un-normalized) Williamson TC2, conservation error, energy Log 2 of ratio of L 2 errors with doubling of resolution 10 -5 10 -5 5.0 ∆ x =768km , ∆ t =150s ∆ x =768km , ∆ t =150s ∆ x =768km , ∆ t =300s ∆ x =768km , ∆ t =300s 10 -6 4.5 ∆ x =384km , ∆ t =300s ∆ x =384km , ∆ t =300s log 2 ( | L 2 (∆ x ) | / | L 2 (∆ x/ 2) | ) 10 -7 4.0 10 -6 10 -8 3.5 10 -9 3.0 ω h + f h , ( N e =4) / ( N e =8) ~ u h , ( N e =4) / ( N e =8) h h , ( N e =4) / ( N e =8) 10 -10 2.5 ω h + f h , ( N e =8) / ( N e =16) ~ u h , ( N e =8) / ( N e =16) h h , ( N e =8) / ( N e =16) 10 -7 10 -11 2.0 0 2 4 6 8 10 0 2 4 6 8 10 0 1 2 3 4 5 time (days) Figure: Vorticity field for the Galewsky test case (day 7), inviscid solution with exact energy conservation (left) and with viscosity (right). Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation

The 3D compressible Euler equations (continuous form) ∂ u ∂ t = − ( ω + fe z ) × u − ∇ ( 1 2 u 2 + gz ) − θ ∇ Π ∂ρ ∂ t = −∇ · ( ρ u ) ∂ Θ ∂ t = −∇ · ( ρθ u ) , Θ = ρθ � R / c v � R Θ Π = c p , c p = R + c v p 0 Energy is defined as � 1 2 ρ u 2 + ρ gz + c v H = ΘΠ d Ω c p Setting δ H δ H δρ = 1 δ H 2 u 2 + gz = Φ , δ u = ρ u = U , δ Θ = Π gives − 1       u ρ ( ω + fe z ) × −∇ − θ ∇ U ∂  = ρ Φ −∇· 0 0      ∂ t Θ Π −∇ · ( θ · ) 0 0 Dave Lee Mixed Mimetic Spectral Elements for Atmospheric Simulation