Discontinuous Galerkin Method

Francis X. GiraldoNaval Research LaboratoryMonterey, CA 93943-5502

www.nrlmry.navy.mil/~giraldo/main.html

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)

• 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

• 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

• 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

• 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

• 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

∑=

=

• 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 *

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

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)(

,

Left Right

Continuous vs. Discontinuous EBG Methods(Stencil)

Continuous EBG Methods (Global Grid Points due to C0)

Γ⋅−=Ω

⋅∇−

∂∂

∫∫ ΓΩdFndF

tq

NNN *ψψψ

Discontinuous EBG Methods (Local Grid Points)

Γ⋅−=Ω

⋅∇−

∂∂

∫∫ ΓΩdFndF

tq

NNN *ψψψ

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

The Power of DG is due to the Flux Integral

Advantage of the DG Method

Γ⋅−=Ω

−⋅∇−

∂∂

∫∫ ΓΩdFndqSF

tq

NNNN *)( ψψψψ

Flux Integral is the Key for 2 Reasons:– It is the only means by which elements communicate (truly

local)

Advantage of the DG Method

CG Communication Stencil

DG Communication Stencil

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).

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

• 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

• 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=

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

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

Applications of the DG Method

• Shallow Water Equations on the Sphere• Mesoscale Atmospheric Model• Coastal Ocean Model

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)

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

Shallow Water Equations on the Sphere (Riemann problem)

Height V Velocity

U Velocity

Applications of the DG Method

• Shallow Water Equations on the Sphere• Mesoscale Atmospheric Model• Coastal Ocean Model

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),(

∂∂

∂∂

=∇

Mesoscale Atmospheric Model (Rising Thermal Bubble)

Potential TemperatureGrid

Applications of the DG Method

• Shallow Water Equations on the Sphere• Mesoscale Atmospheric Model• Coastal Ocean Model

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∇•∇=•∇

Coastal Ocean Model(Rossby Soliton Waves in a Channel)

Grid Geopotential Height