discontinuous galerkin method - cosmo model · giraldo_discont_galerkin_method author: abierman...
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![Page 1: Discontinuous Galerkin Method - COSMO model · giraldo_discont_galerkin_method Author: abierman Created Date: 4/4/2006 3:28:24 PM](https://reader035.vdocuments.us/reader035/viewer/2022071018/5fd1b7991249f77eec7ef738/html5/thumbnails/1.jpg)
Discontinuous Galerkin Method
Francis X. GiraldoNaval Research LaboratoryMonterey, CA 93943-5502
www.nrlmry.navy.mil/~giraldo/main.html
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Basis Function Choices for EBG Methods
• Modal Space
– Spectral (amplitude-frequency) space
– Orthogonal functions (e.g., Legendre, PKD)
• Nodal Space
– Physical variable space
– Cardinal functions (e.g., Lagrange polynomials)
• Continuous
– FE and SE Methods (local)
– Spectral Transform Method (global)
• Discontinuous
– FV and DG Methods (local)
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• High order like spherical harmonics yet local like finite elementmethods– globally conservative = good for hydrostatic primitive equation
• Theoretically optimal for self-adjoint operators (elliptic equations)– Excellent for incompressible Navier-Stokes– also extremely good for non-self-adjoint operators (hyperbolic
equations)• Simple to construct efficient semi-implicit time-integrators
– Semi-implicit = nonlinear terms are explicit and linear terms areimplicit
Properties of Continuous EBG Methods
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• High order like Continuous Galerkin• Completely local in nature
– no global assembly/DSS required as in CG (truly local)• High order generalization of the FV
– locally and globally conservative = excellent choice for non-hydrostatic equations (mesoscale models)
– upwinding implemented naturally (via Riemann solvers)• Designed for hyperbolic equations (i.e., shock waves)
– Simple provision for elliptic equations• Not so straightforward to construct efficient semi-implicit time-
integrators (inherently nonlinear)
Properties of Discontinuous EBG Methods
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• Primitive Equations:
• Approximate the solution as:
• Write Primitive Equations as:
• Weak Problem Statement: Find– (which assumes C0)
– such that
– where
( ) 0)( =−∂
∂≡ N
NN qS
tq
qR
11 )( HHqN ∈∀Ω∈ ψ
( ) 0=Ω∫ΩdqR Nψ
e
N
e
e
Ω=Ω=U
1
Discretization by ContinuousContinuous EBG Methods
)(qStq
=∂∂
i
M
iiN qzyxzyxq
N
∑=
=1
),,(),,( ψ
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• Primitive Equations:
• Approximate the solution as:
• Write Primitive Equations as:
• Weak Problem Statement: Find– (which does not assume C0)
– such that
– where
( ) 0)( =−∂
∂≡ N
NN qS
tq
qR
( ) 0=Ω∫ΩdqR Nψ
e
N
e
e
Ω=Ω=U
1
Discretization by Discontinuouscontinuous EBG Methods
)(qStq
=∂∂
i
M
iiN qzyxzyxq
N
∑=
=1
),,(),,( ψ
22 )( LLqN ∈∀Ω∈ ψ
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• Conservation Law:
• Weak Form:
• Approximation:
• Integration By Parts:
0=Ω
⋅∇+
∂∂
∫ΩdF
tq
ψ
0=⋅∇+∂∂
Ftq
( ) 0=Ω
⋅∇−⋅∇+
∂∂
∫ΩdFF
tq
NNN ψψψ
Difference between CG and DG
( ) ( ) jj
M
jN qq
N
ηξψηξ ,,1
∑=
=
( ) ( ) jj
M
jN FF
N
ηξψηξ ,,1
∑=
=
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• Using Divergence Theorem and C0 continuity of CG:
• Using Divergence Theorem (DG):
• Constant ψ for DG yields FV:
Γ⋅−=Ω
⋅∇−
∂∂
∫∫ ΓΩdFndF
tq
NNN *ψψψ
Γ⋅−=Ω
⋅∇−
∂∂
∫∫ ΓΩdFndF
tq
NNN *ψψψ
Difference between CG and DG
Γ⋅−=Ω∂∂
∫∫ ΓΩdFnd
tq *
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Continuous EBG Methods(Matrix Form)
Global Matrix Form
Element-wise Approximation
( ) 0,, =+∂
∂J
TJI
JJI FD
tq
M
ejie
ji dMe
Ω= ∫Ωψψ)(
,Element Matrices ejie
ji dDe
Ω∇= ∫Ωψψ)(
,
)(
0
)( ej
N
jj
eN qq ∑
=
= ψ )(
0
)( ej
N
jj
eN FF ∑
=
= ψ
DSS)(
,1,e
ji
N
eJI MME
=Λ=
)(,
1,
eji
N
eJI DD
E
=Λ=
Domain Decompositione
N
e
E
Ω=Ω=U
1
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Discontinuous EBG Methods(Matrix Form)
Local Matrix Form
Element-wise Approximation
( ) ( ) ( )(*))()(,
)()(,
)()(
, je
jTe
jie
jTe
ji
eje
ji FFMFDt
qM −=+
∂∂
ejie
ji dMe
Ω= ∫Ωψψ)(
,Element Matrices ejie
ji dDe
Ω∇= ∫Ωψψ)(
,
)(
0
)( ej
N
jj
eN qq ∑
=
= ψ )(
0
)( ej
N
jj
eN FF ∑
=
= ψ
Domain Decompositione
N
e
E
Ω=Ω=U
1
ejie
ji dnMe
Γ= ∫Γψψ)(
,
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Left Right
Continuous vs. Discontinuous EBG Methods(Stencil)
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Continuous EBG Methods (Global Grid Points due to C0)
Γ⋅−=Ω
⋅∇−
∂∂
∫∫ ΓΩdFndF
tq
NNN *ψψψ
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Discontinuous EBG Methods (Local Grid Points)
Γ⋅−=Ω
⋅∇−
∂∂
∫∫ ΓΩdFndF
tq
NNN *ψψψ
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I-1 I I+1
Left Right
I-1 I I+1
Left Right
CG
DG
Difference between CG and DG
Γ⋅−=Ω
⋅∇−
∂∂
∫∫ ΓΩdFndF
tq
NNN *ψψψ
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The Power of DG is due to the Flux Integral
Advantage of the DG Method
Γ⋅−=Ω
−⋅∇−
∂∂
∫∫ ΓΩdFndqSF
tq
NNNN *)( ψψψψ
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Flux Integral is the Key for 2 Reasons:– It is the only means by which elements communicate (truly
local)
Advantage of the DG Method
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CG Communication Stencil
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DG Communication Stencil
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Stencil of CG and DG Methods
The DG communication stencil of the element T is given by the solid triangles (3 triangles)whereas the CG stencil of the element T is given by the solid and dashed triangles (triangles).
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Flux Integral is the Key for 2 Reasons:– It is the only means by which elements communicate (truly
local)– It allows a way to introduce semi-analytic methods (Riemann
Solvers)
Advantage of the DG Method
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• Problem Statement
• Name of the Game is to approximate:
– Where R (for Rotational Invariant PDEs) maps:
Advantage of the DG Method(Numerical Flux and 3D to 1D Mapping)
)(ˆ *1*NN RqFRFn −=⋅
Twvuq ),,,( φφφφ= Trtn uuuq ),,,(ˆ φφφφ=
Γ⋅−=Ω
−⋅∇−
∂∂
∫∫ ΓΩdFndqSF
tq
NNNN *)( ψψψψ
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• Using the 1D Rotation Mapping we solve the Riemann Problem for:
• The Associated Riemann Problem is:
• Where F* is selected to best approximate the exact solution (e.g.,Godunov, Osher, Roe, Rusanov, HLLC)
Advantage of the DG Method(Numerical Flux and Riemann Problem)
Trtn uuuq ),,,(ˆ φφφφ=
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0 90 180 270 360
0
50
100
λ
u(λ
)
These results are for a scale contraction problem (passive advection of a discontinuous functionof fluid). The mass profile along the Equator are shown for the CG and DG methods using N=8polynomials. Even with strong spatial filtering, the CG method experiences Gibbs phenomenawhile the DG method only feels slight oscillations.
Advantage of the DG Method(solid body rotation of circular column on sphere)
0 90 180 270 360
0
50
100
λu(
λ)
CG DG
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These results are for a shock wave dynamics problem (cylindrical shock wave on a stationarysphere). The solution profile along the Equator are shown for the CG (left panel) and DG (rightpanel) methods using N=1 polynomials. Even with strong spatial filtering, the CG methodexperiences Gibbs phenomena while the DG method only feels a small undershoot at the base ofthe shock fronts.
Advantage of the DG Method(Riemann problem on sphere for SWE)
CG DG
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Applications of the DG Method
• Shallow Water Equations on the Sphere• Mesoscale Atmospheric Model• Coastal Ocean Model
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Shallow Water Equations on the Sphere(Governing Equations)
( ) 0=•∇+∂∂
ut
ϕϕ
(Mass)
(Momentum)
,),,( Twvuu =
,),,( Tzyxx =
T
zyx),,(
∂∂
∂∂
∂∂
=∇
0)(2
21
232 =×
Ω+
+⊗•∇+
∂∂
uka
zIuu
tu
ϕϕϕϕ
gh=ϕ (Geopotential)
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Shallow Water Equations on the Sphere(Williamson et al. Case 2)
These results show the exponential (spectral) convergence of the SE and DG - that is, the errordecreases exponentially with the order N of the basis function
( )1error +∆∝ NxO
Quadrilaterals Triangles
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Shallow Water Equations on the Sphere (Riemann problem)
Height V Velocity
U Velocity
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Applications of the DG Method
• Shallow Water Equations on the Sphere• Mesoscale Atmospheric Model• Coastal Ocean Model
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Mesoscale Atmospheric Model(Governing Equations)
( ) 0=•∇+∂∂
ut
ρρ
(Mass)
(Momentum)
(Energy)
( ) ( ) kguxkfIPuutu
ρρρρ
−−=+⊗•∇+∂
∂3
(Equation of State )
( ) 0=•∇+∂
∂u
tθρ
θρ
vp cc
PR
PP/
00
=
θρ
(Definitions )π
θT
=
,),( Twuu =
,),( Tzxx =
T
zx),(
∂∂
∂∂
=∇
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Mesoscale Atmospheric Model (Rising Thermal Bubble)
Potential TemperatureGrid
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Applications of the DG Method
• Shallow Water Equations on the Sphere• Mesoscale Atmospheric Model• Coastal Ocean Model
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Coastal Ocean Model(Governing Equations)
( ) 0=•∇+∂∂
ut
ϕϕ (Mass)
(Momentum)
,),( Tvuu =
,),( Tyxx =
T
yx),(
∂∂
∂∂
=∇
uguka
zqIuu
tu
γϕρτ
ϕϕϕϕ
−=×Ω
+
−+⊗•∇+
∂∂
)(2
21
222
gh=ϕ (Geopotential)
(Temporary Variable:
Such that)
uq ϕν∇=
( )uq ϕν∇•∇=•∇
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Coastal Ocean Model(Rossby Soliton Waves in a Channel)
Grid Geopotential Height