1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Rocky Mountain Mathematics Consortium Summer School
Conservation Laws & Applications
Lecture V: Discontinuous Galerkin Methods
James A. Rossmanith
Department of Mathematics
University of Wisconsin – Madison
July 2st , 2010
J.A. Rossmanith | RMMC 2010
1/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Outline
1 1D DG-FEM
2 2D Cartesian DG-FEM
3 2D mesh generation
4 2D unstructured DG-FEM
J.A. Rossmanith | RMMC 2010
2/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Outline
1 1D DG-FEM
2 2D Cartesian DG-FEM
3 2D mesh generation
4 2D unstructured DG-FEM
J.A. Rossmanith | RMMC 2010
3/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Basic idea
Spatial discretization [Cockburn & Shu, 1990’s]
Map each element to [−1, 1]:
x = xi + ξ
∆x
2
Basis functions: 
√
√
ff
√
´
´
5` 2
7` 3
φ(`) (ξ) = 1, 3 ξ,
3ξ − 1 ,
5ξ − 3ξ , . . .
2
2
Orthogonality:
1
2
Z
1
φ(`) (ξ) φ(k) (ξ) dξ = δ`k
−1
Galerkin expansion:
n
˛
X
˛
q h (t, x)˛ =
Q(`) (t) φ(`) (ξ(x))
T
J.A. Rossmanith | RMMC 2010
`=1
4/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Basic idea
Cofficients are computed via L2 -projection:
#2
Z "
n
“
”
X
1 1
E Q(1) (t), . . . , Q(n) (t) =
Q(`) (t) φ(`) (ξ) dξ
q(t, ξ) −
2 −1
`=1
∂E
=
∂Q(k)
1
Z
"
q(t, ξ) −
−1
∴
J.A. Rossmanith | RMMC 2010
n
X
#
Q(`) (t) φ(`) (ξ) φ(k) (ξ) dξ = 0
`=1
Q(k) (t) =
1
2
Z
1
q(t, ξ) φ(k) (ξ) dξ
−1
5/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Basic idea
Weak formulation:
Z
o
1 1 (`) n
φ
q,t + f (q),x dξ = 0
2 −1
 Z 1
ff
Z 1
d 1
1
φ(`) q dξ = −
φ(`) f (q),ξ dξ
dt 2 −1
∆x −1
n
X
q(t, ξ) ∼
Q(k) (t) φ(k) (ξ)
k=1
(k)
Q̇i
1
=
∆x
Z
1
„
»
–
˛ «
1
˛
(k)
φ,ξ (ξ) dξ −
f q h (t, ξ)˛
φ(k) (1) Fi+ 1 − φ(k) (−1) Fi− 1
2
2
∆x
Ti
−1
Conservation:
k=1
Time integrate
J.A. Rossmanith | RMMC 2010
(1)
=⇒
=⇒
Q̇i
h
(1)
Qi
»
–
1
Fi+ 1 − Fi− 1
2
2
∆x
»
–
h
in
∆t
(1)
= Qi
−
F̄i+ 1 − F̄i− 1
2
2
∆x
=−
in+1
6/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Basic idea
Weak formulation:
Z
o
1 1 (`) n
φ
q,t + f (q),x dξ = 0
2 −1
 Z 1
ff
Z 1
d 1
1
φ(`) q dξ = −
φ(`) f (q),ξ dξ
dt 2 −1
∆x −1
n
X
q(t, ξ) ∼
Q(k) (t) φ(k) (ξ)
k=1
(k)
Q̇i
1
=
∆x
Z
1
„
»
–
˛ «
1
˛
(k)
φ,ξ (ξ) dξ −
f q h (t, ξ)˛
φ(k) (1) Fi+ 1 − φ(k) (−1) Fi− 1
2
2
∆x
Ti
−1
Conservation:
k=1
Time integrate
J.A. Rossmanith | RMMC 2010
(1)
=⇒
=⇒
Q̇i
h
(1)
Qi
»
–
1
Fi+ 1 − Fi− 1
2
2
∆x
»
–
h
in
∆t
(1)
= Qi
−
F̄i+ 1 − F̄i− 1
2
2
∆x
=−
in+1
6/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Basic idea
Time-stepping:
Space-time methods (implicit)
Lax-Wendroff (Taylor-series + replace time by spatial derivatives)
Total variation diminishing Runge-Kutta:
U ? = U n + ∆t L(U n )
3
1
1
U ?? = U n + U ? + ∆t L(U ? )
4
4
4
1 n 2 ?? 2
n+1
U
= U + U + ∆t L(U ?? )
3
3
3
Numerical fluxes:
Can use either exact or approximate Riemann solvers
Unlike WP-FVM, order is not tied to Riemann solver
Only need F (q̄, q̄) = f (q̄)
Local Lax-Friedrichs:
1
1
F (Qr , Q` ) = (f (Q` ) + f (Qr )) − s (Qr − Q` ) ,
2
2
J.A. Rossmanith | RMMC 2010
˛
˛
s = max˛λ(f,q )˛
7/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Basic idea
Time-stepping:
Space-time methods (implicit)
Lax-Wendroff (Taylor-series + replace time by spatial derivatives)
Total variation diminishing Runge-Kutta:
U ? = U n + ∆t L(U n )
3
1
1
U ?? = U n + U ? + ∆t L(U ? )
4
4
4
1 n 2 ?? 2
n+1
U
= U + U + ∆t L(U ?? )
3
3
3
Numerical fluxes:
Can use either exact or approximate Riemann solvers
Unlike WP-FVM, order is not tied to Riemann solver
Only need F (q̄, q̄) = f (q̄)
Local Lax-Friedrichs:
1
1
F (Qr , Q` ) = (f (Q` ) + f (Qr )) − s (Qr − Q` ) ,
2
2
J.A. Rossmanith | RMMC 2010
˛
˛
s = max˛λ(f,q )˛
7/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Gaussian quadrature
Need to evaluate integrals of the form
Z 1
I=
f (ξ) dξ
−1
Gaussian quadrature has maximum degree of precision: 2n − 1
Gaussian quadrature rules:
I1 = 2f (0)
X „ 1 «
I2 =
f ±√
3
±
„ √ «
8
5 X
3
I3 = f (0) +
f ±√
9
9 ±
5
When integrating against φ(`) need rule with N points
(`)
When integrating against φ,ξ need rule with N − 1 points
J.A. Rossmanith | RMMC 2010
8/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Moment limiters
[Krivodonova, 2007]
Moment representation:
Q(1) , Q(2) , Q(3) , . . .
Moment limiter:
“
”
“
””
“
(k)
(k)
(k−1)
(k−1)
(k−1)
(k−1)
Qi = minmod Qi , c(k) Qi
− Qi−1
, c(k) Qi+1 − Qi
..
.
(2)
Qi
“
“
”
“
””
(2)
(1)
(1)
(1)
(1)
= minmod Qi , c(2) Qi − Qi−1 , c(2) Qi+1 − Qi
J.A. Rossmanith | RMMC 2010
9/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Artificial viscosity
[Persson & Peraire, 2006]
q,t + f (q),x = (ε B(q) v),x
v = q,x
Cell-by-cell smoothness indicator:
qh =
N
X
Q(`) φ(`) ,
q̂ h =
`=1
hq h − q̂ h , q h − q̂ h i
,
hq h , q h i
(
0
if S < TOL
ε :=
ε0
if S ≥ TOL
S :=
N
−1
X
Q(`) φ(`)
`=1
S ∼ ∆x2N (smooth solutions)
Local DG method for elliptic & parabolic PDEs:
Multiply both equations by same test function, φ(`) , and 1 int-by-parts
Difficulties: time-step restriction due to parabolic part, parameters
J.A. Rossmanith | RMMC 2010
10/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Discontinuous Galerkin FEM
Why?
Comparison of properties:
High-order
Unstructured meshes
Stability for conservation laws
Explicit time-stepping
Elliptic equations
FVM
8
4
4
4
4
FDM
4
8
4
4
4
FEM
4
4
8
(4)
4
DG
4
4
4
4
(4)
Difficulties with DG-FEM
No universally accepted limiters
Generally small time-restrictions due to sub-grid resolution
Errors in replacing exact integrals by Gaussian quadrature
Entropy stability and explicit time-stepping
J.A. Rossmanith | RMMC 2010
11/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Discontinuous Galerkin FEM
A brief history
First introduced by [Reed & Hill, 1973] for neutron transport:
σq + ∇ · (uq) = f
First analysis by [Lesaint & Raviart, 1974] showing O(hN )
1
Sharp analysis [Johnson, 1986] showing O(hN + 2 )
1990’s: Extension to general conservation laws [Cockburn & Shu]
Out of all this came RKDG (Runge-Kutta discontinuous Galerkin)
Last 15 years: high-order, adaptivity, nodal/modal, . . .:
Hesthaven, Warburton, Karniadakis, Kopriva, Shu, Persson, Munz, . . .
Applications:
Fluid dynamics
Aero-acoustics
Maxwell equations
Plasma physics
Shallow water waves
J.A. Rossmanith | RMMC 2010
12/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Outline
1 1D DG-FEM
2 2D Cartesian DG-FEM
3 2D mesh generation
4 2D unstructured DG-FEM
J.A. Rossmanith | RMMC 2010
13/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
DG-FEM on Cartesian grids
Map each element to canonical element [−1, 1] × [−1, 1]
x = xi + ξ
∆x
2
y = yj + η
∆y
2
Basis functions (up to third-order accuracy):
√
√
n √
√
´
´o
5` 2
5` 2
3 ξ,
3 η, 3 ξη,
3ξ − 1 ,
3η − 1
φ(`) = 1,
2
2
Orthonormality:
D
Z Z
E
1 1 1 (m)
φ(m) , φ(n) :=
φ (ξ, η) φ(n) (ξ, η) dξ dη = δmn
4 −1 −1
Galerkin approximation:
6
X
˛
(k)
q h (t, ξ, η)˛T ≡
Qij (t) φ(k) (ξ, η)
ij
k=1
J.A. Rossmanith | RMMC 2010
14/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
DG-FEM on Cartesian grids
Semi-discrete form:
(`)
(`)
∆Fij
∆Gij
d (`)
(`)
Qij = Nij −
−
dt
∆x
∆y
Internal and flux difference parts in the semi-discrete form:
–
Z Z »
1 1 1
1 (`)
1 (`)
(`)
Nij =
φ,ξ f (q h ) +
φ,η g(q h ) dξ dη,
2 −1 −1 ∆x
∆y
» Z 1
–ξ=1
1
(`)
∆Fij =
φ(`) f (q h ) dη
2 −1
ξ=−1
» Z 1
–η=1
1
(`)
∆Gij =
φ(`) g(q h ) dξ
2 −1
η=−1
Integrals approximated via Gaussian quadrature rules
Surface integrals: 1D GQ with N points
Volume integrals: 2D tensor product GQ with (N − 1) × (N − 1) points
J.A. Rossmanith | RMMC 2010
15/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Moment limiters
[Krivodonova, 2007]
In multi-D, moment limiters become more complicated
For example, with a third-order scheme:
Q(4) ξ 2
Q(5) ξη
Q(2) ξ
Q(6) η 2
Q(3) η
Q(1)
Limit highest-order first:
(5)
Qij
“
”
(4)
(4)
(2)
− (2)
Qij = minmod Qij , ∆+
x Qij , ∆x Qij
“
”
(5)
(2)
− (2)
+ (3)
− (3)
= minmod Qij , ∆+
y Qij , ∆y Qij , ∆x Qij , ∆x Qij
“
”
(6)
(6)
(3)
− (3)
Qij = minmod Qij , ∆+
y Qij , ∆y Qij
If anything happened, go to next order:
“
”
(2)
(2)
(1)
− (1)
Qij = minmod Qij , ∆+
x Qij , ∆x Qij
“
”
(3)
(3)
(1)
− (1)
Qij = minmod Qij , ∆+
y Qij , ∆y Qij
J.A. Rossmanith | RMMC 2010
16/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Artificial viscosity
[Persson & Peraire, 2006]
q,t + ∇ · F(q) = ∇ · (ε B(q) v)
v = ∇q
Cell-by-cell smoothness indicator:
K(N −1)
K(N )
qh =
X
Q(`) φ(`) ,
q̂ h =
`=1
hq h − q̂ h , q h − q̂ h i
,
hq h , q h i
(
0
if S < TOL
ε :=
ε0
if S ≥ TOL
S :=
X
Q(`) φ(`)
`=1
S ∼ h2N (smooth solutions)
Local DG method for elliptic & parabolic PDEs:
Multiply each equation by same test function, φ(`) , and 1 int-by-parts
Difficulties: time-step restriction due to parabolic part, parameters
J.A. Rossmanith | RMMC 2010
17/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Outline
1 1D DG-FEM
2 2D Cartesian DG-FEM
3 2D mesh generation
4 2D unstructured DG-FEM
J.A. Rossmanith | RMMC 2010
18/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Triangular elements
p(i,1:2) := x and y coordinates of ith node
t(j,1:3) := list of 3 node indeces that make up element j
All other info about nodes, elements, & edges can be derived from this
Example:
t(5,1) = 7,
J.A. Rossmanith | RMMC 2010
t(5,2) = 25,
t(5,3) = 18
19/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given points, how do we form a triangulation?
Mesh quality
Minimum angle:
„
θT := min
p=1,2,3
Delaunay:
θTp
60◦
«
,
τh := min θT ,
T
0 ≤ τh ≤ 1
for fixed point locations, this triangulation maximizes τh
Delaunay triangulation
Each point gets region of influence based on distance: Voronoi diagram
Hpq := {x : kp − xk ≤ kq − xk}
=⇒
regionp = ∩ Hpq
This diagram is unique, area of overlap is zero, partitions R
∀q
2
Delaunay triangulation is dual to Voronoi: only connect if share edge
Voronoi diagram is unique, Delaunay tri. is unique up to degeneracies
Degeneracy:
4+ points are co-circular =⇒ resolve with coin flip
J.A. Rossmanith | RMMC 2010
20/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given points, how do we form a triangulation?
Delaunay triangulation
J.A. Rossmanith | RMMC 2010
21/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given points, how do we form a triangulation?
Delaunay triangulation
J.A. Rossmanith | RMMC 2010
22/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given points, how do we form a triangulation?
Delaunay triangulation
J.A. Rossmanith | RMMC 2010
23/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given points, how do we form a triangulation?
Delaunay triangulation
J.A. Rossmanith | RMMC 2010
24/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given ∂Ω, how do we form a triangulation?
Mesh generator of [Persson & Strang, 2004]
Define boundary via
8
>
<−
Φ(x) = +
>
:
0
signed distance function:
if
if
if
x∈Ω
x∈
/Ω ,
x ∈ ∂Ω
k∇Φ(x)k2 = 1
∀ x ∈ R2
Start with equi-distributed points in Ω and triangulate
J.A. Rossmanith | RMMC 2010
25/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Mesh generation
Given ∂Ω, how do we form a triangulation?
Mesh generator of [Persson & Strang, 2004]
Spring dynamics: spring const. ∝ desired h, only allow repulsion
Points that leave Ω are projected onto ∂Ω via
x ← x − Φ(x) ∇Φ(x)
Rinse and repeat, occasionally retriangulating
J.A. Rossmanith | RMMC 2010
26/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Outline
1 1D DG-FEM
2 2D Cartesian DG-FEM
3 2D mesh generation
4 2D unstructured DG-FEM
J.A. Rossmanith | RMMC 2010
27/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
DG-FEM on unstructured grids
Map each element to canonical triangle
„
«
„
«
„
«
1
1
2
1
1 2
,−
− ,−
↔
↔ − ,
3
3
3
3
3 3
1
x = (x1 + x2 + x3 ) + ξ (x2 − x1 ) + η (x3 − x1 )
3
1
y = (y1 + y2 + y3 ) + ξ (y2 − y1 ) + η (y3 − y1 )
3
Basis functions (before Gram-Schmidt orthonormalization):
˘
¯
φ(`) = 1, ξ, η, ξ 2 , ξη, η 2 , ξ 3 , ξ 2 η, ξη 2 , η 3 , . . .
J.A. Rossmanith | RMMC 2010
28/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
DG-FEM on unstructured grids
Galerkin expansion:
k(k+1)/2
q h (t, x) =
X
Q(`) (t) φ(`) (x)
n=1
∀T start with q,t + ∇ · F(q) = 0 and obtain semi-discrete weak-form:
Z
I
d (`)
1
1
Q =
∇φ(`) · F(q) dx −
φ(`) F(q) · ds
dt
|T | T
|T | ∂T
|
{z
} |
{z
}
Interior
Edge
Interior: Gauss quad, Edge: approx Riemann soln, then quadrature
J.A. Rossmanith | RMMC 2010
29/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Gaussian quadrature
[D.A. Dunavant, 1985]
J.A. Rossmanith | RMMC 2010
30/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Artificial viscosity
[Persson & Peraire, 2006]
q,t + ∇ · F(q) = ∇ · (ε B(q) v)
v = ∇q
Cell-by-cell smoothness indicator:
K(N −1)
K(N )
qh =
X
Q(`) φ(`) ,
q̂ h =
`=1
hq h − q̂ h , q h − q̂ h i
,
hq h , q h i
(
0
if S < TOL
ε :=
ε0
if S ≥ TOL
S :=
X
Q(`) φ(`)
`=1
S ∼ h2N (smooth solutions)
Local DG method for elliptic & parabolic PDEs:
Multiply each equation by same test function, φ(`) , and 1 int-by-parts
Difficulties: time-step restriction due to parabolic part, parameters
J.A. Rossmanith | RMMC 2010
31/35
1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Space/time methods
Basic idea
Take space-time Galerkin ansatz:
N
˛
X
˛
q h (t, x)˛ =
Q(k) φ(k) (t, x)
T
k=1
Weak formulation:
1
|T |
Z tn+1 I
Q(`), n+1 = ±Q(`), n +
1
−
|T |
J.A. Rossmanith | RMMC 2010
tn
Z
tn+1
tn
Z h
i
(`)
φ,t q + ∇φ(`) · F(q) dx dt
T
φ(`) F(q) · ds dt
∂T
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1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Space/time methods
Entropy variables
Space/time DG-FEM in entropy variables:
Find vh ∈ V h such that
BDG (vh , wh ) = 0 , ∀ wh ∈ V h
with
BDG (v, w)
=
N
−1
X
XZ Z
n=0
+
K∈T
XZ Z
K∈T
+
In
In K
−(q(v) · w,t + fi (v) · w,xi ) dx dt
w(x− ) · h(v(x− ), v(x+ ); n) ds dt
∂K
!
XZ `
n+1
n
n ´
w(tn+1
)
·
q(v(t
))
−
w(t
)
·
q(v(t
))
dx
+
−
−
−
K∈T
K
Entropy stability:
Z
Z
U (q(vh (tN
,
x)))
dx
≤
U (q(vh (t0− , x)) dx
−
Ω
J.A. Rossmanith | RMMC 2010
Ω
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1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Lax-Wendroff time-stepping
Basic idea
Another alternative to RK is to do a Taylor series in time
This turns out to be useful for local time-stepping
Taylor series expansion:
q,t + ∇ · F(q) = ψ(q)
q(t + ∆t, x, y) = q + ∆t q,t +
ˆ
˜
∆t2
q,tt + O(∆t3 ) = q − ∆t ∇ · F − Ψ
2
∆t
F,q · {∇ · F − ψ} + O(∆t2 )
2
∆t
Ψ := ψ −
ψ,q · {∇ · F − ψ} + O(∆t2 ).
2
Weak formulation:
ZZ
n
on+1 n
on
∆t
(`)
(`)
∇φ(`) · F n dx
Qi
= Qi
+
|Ti |
Ti
I
ZZ
∆t
∆t
−
φ(`) Fn · n ds +
φ(`) Ψn dx
|Ti | ∂Ti
|Ti |
Ti
F := F −
J.A. Rossmanith | RMMC 2010
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1D DG-FEM
2D Cartesian DG-FEM
2D mesh generation
2D unstructured DG-FEM
Lax-Wendroff time-stepping
Asynchronous steps
Each element has its own time-step:
1 Sort each element by their current time ti
2 Update the element that is lagging furthest behind
3 Apply flux corrections to maintain global conservation
4 Place the updated element in the correct location of the ordered list
5 Rinse and repeat
J.A. Rossmanith | RMMC 2010
35/35