Discontinuous Galerkin Methods
Li, Yang
FerienAkademie 2008
Contents
Li, Yang
Methods of solving PDEs
Introduction of DG Methods
Working with 1-Dimension
FerienAkademie 2008
Methods of solving PDEs
Li, Yang
FerienAkademie 2008
Finite Difference Method
Finite Volume Method
Finite Element Method
PDEs
Methods of solving PDEs
Li, Yang
FerienAkademie 2008
E.g. 1D scalar conservation law
u f

 g, x  
t x
with initial conditions and boundary conditions
on the boundary 
How to get the
unknown solution u ( x, t )
approximate solution
uh ( x, t ) ?
flux f (u )
Is it satisfied the
equation?
prescribed force g ( x, t )
Methods of solving PDEs
Li, Yang
FerienAkademie 2008
Finite Difference Method
k 1
duh ( x , t ) f h ( x , t )  f h ( x

k
k 1
dt
h h
k
k 1
, t)
 g ( xk , t )
a grid xk , k  1...N p
local grid size h k  x k 1  x k
assume:
2
2
i 0
i 0
x  [ x k 1 , x k 1 ] : uh ( x, t )   ai (t )( x  x k )i , f h ( x, t )   bi (t )( x  x k )i
uh f h

 g ( x, t )
Residual: Rh ( x, t ) 
t
x
Methods of solving PDEs
Li, Yang
Finite Difference
Method

Simple to
implement
FerienAkademie 2008
Element-based
discretization
to ensure
geometry
flexibility

Ill-suited to
deal with
complex
geometries
Methods of solving PDEs
Li, Yang
FerienAkademie 2008
Finite Volume Method
k
k
u
f (u )
x  D : Rh ( x, t ) 

 g ( x, t )
t
x
The actual
k
numerical scheme
1
 ( xk
element staggered grid D  [ x , x will]depend
, x k 1/ 2 upon
2
problem geometry
solution is approximated on the element
by a
and mesh
k
construction.
constant u (t )
Difficult when highk
order
k du
k 1/ 2
k 1/ 2
k k
Divergence Theorem
h
f
f
h g
reconstruction.
k 1/ 2
k
k 1/ 2
dt
Reconstruction of uh ( x) 
Solution uh
p
 a (x  x
i 0
i
k i
)
To find the p+1 unknown
coefficients need
information at least from
p+1 cells
 x k 1 )
Methods of solving PDEs
Li, Yang
FerienAkademie 2008
Finite Element Method
Np
assume the local solution: x  D k : uh ( x)   bn n ( x)
n 1
element Dk  [ x k , x k 1 ] locally defined basis function  n ( x)
global representation of uh :
K
uh ( x )   u ( x k ) N k ( x )
k 1
where N i ( x j )  ijis the basis function.
define a space of test functions,Vh , and require
the residual is orthogonal to all test functions:
uh f h
 (

 g h )h ( x)dx  0, h  Vh
t
x
Methods of solving PDEs
Li, Yang
FerienAkademie 2008
Finite Element Method
Classical choice: the spaces spanned by the basis functions and test
functions are the same.
K
h ( x)   v( x k ) N k ( x)


k 1
Easy to extend to highh  Vh The semi-discrete
Since
the
residual
has
to
vanish
for
all
order approximation by
scheme becomes

u

f
adding additional
( h  h  g h )N j ( x)dximplicit
 0, and M must be
degrees of freedom to t
inverted
x
the element.

du h
M
 Sf h  Mg h
dt
j
dN
Mij   N i ( x) N j ( x)dx, Sij   N i ( x)
dx


dx
Introduction of DG Methods
Li, Yang
FerienAkademie 2008
 The Discontinuous Galerkin method is somewhere between a finite
element and a finite volume method and has many good features of
both, utilizing a space of basis and test functions that mimics the
finite element method but satisfying the equation in a sense closer to
the finite volume method.
 It provides a practical framework for the development of high-order
accurate methods using unstructured grids. The method is well
suited for large-scale time-dependent computations in which high
accuracy is required.
 An important distinction between the DG method and the usual
finite-element method is that in the DG method the resulting
equations are local to the generating element. The solution within
each element is not reconstructed by looking to neighboring
elements. Its compact formulation can be applied near boundaries
without special treatment, which greatly increases the robustness
and accuracy of any boundary condition implementation.
Introduction of DG Methods
Li, Yang
FerienAkademie 2008
From FEM and FVM to DG-FEM
maintain the definition of elements as in the FEM D k  [ x k , x k 1 ]
but new definition of vector of unknowns
u h  [u1 , u 2 , u 2 , u 3 ,
, u K 1 , u K , u K , u K 1 ]T
Assume the local solution in each element is: (likewise for the flux)
k 1
k
1
x

x
x

x
k
k
k
k 1
k i k
x  D : uh ( x)  u k

u

u
li ( x)

k 1
k 1
k
x x
x x
i 0
Define The space of basis functions: Vh  
K
k 1
l 
k 1
i i 0
The local residual is:
k
k

u

f
x  D k : Rh ( x, t )  h  h  g ( x, t ),
t
x
Introduction of DG Methods
Li, Yang
FerienAkademie 2008
Require that the residual is orthogonal to all test functions h  Vh :

D
k
R
(
x
,
t
)
l
h
i ( x)dx  0
k
Similar to FVM, use Gauss’ theorem:
k

j
uhk k
j
k
k
k k x k 1
l

f

gl
dx


[
f
h l j ]x k
Dk t j h x j
introduce the numerical flux, f , as the unique value to be used at
the interface and obtained by coming information from both elements.

Dk
k

j
k 1
uhk k
l j  f hk j  gl kj dx  [ f l kj ]xxk
t
x
Weak
Form
applying Gauss’ theorem again:

Dk

k x k 1
j xk
Rh ( x, t )l ( x)dx  [( f  f )l ]
k
j
k
h
Strong
Form
Introduction of DG Methods
Li, Yang
FerienAkademie 2008
More general form
Consider the nonlinear, scalar, conservation law:
u f (u )

 0, x  [ L, R]
t
x
subject to appropriate initial conditions u ( x, 0)  u0 ( x).
The boundary conditions are provided when the boundary is an
inflow boundary:
u ( L, t )  g1 (t ) when f h (u ( L, t ))  0,
u ( R, t )  g 2 (t ) when f h (u ( R, t ))  0,
We still assume that the global solution can be well approximated by
a space of piecewise polynomial functions, defined on the union
of D k , and require the residual to be orthogonal to space of the test
K
k
functions, h   k 1h  Vh
Introduction of DG Methods
Li, Yang
FerienAkademie 2008
recover the locally defined weak formulation:
k
uhk k
d

k
k
 k
h
ˆ
(


f
(
u
)
)
dx


n

f
Dk t h h h dx
 Dk h dx,
and the strong form:
uhk f hk (uhk ) k
Dk ( t  x )h dx 

D
k
k

k
ˆ
n

(
f
(
u
)

f
)

h
h
h dx,
k
Assume that all local test functions can be represented by using a
local polynomial basis, n ( x) , as
Np
x  D k : hk ( x)   ˆnk n ( x)
and leads to N p equations as:
n 1
uhk
k
k d n

ˆ
(


f
(
u
)
)
dx


n

f
Dk t n h h dx
Dk  n dx,
uhk f hk (uhk )
Dk ( t  x ) n dx 

D
k
k

ˆ
n

(
f
(
u
)

f
) n dx,
h
h
k
Working with 1-Dimension
Li, Yang
FerienAkademie 2008
E.g.
 Choose the basis functions: Jacobi polynomials
 Integral: Gaussian quadrature
 Time: 4th order explicit RK method
 Simple algorithm steps:
Generate simple mesh
Construct the matrices
Solve the equation system
Working with 1-Dimension
Li, Yang
FerienAkademie 2008
Working with 1-Dimension
Li, Yang
FerienAkademie 2008
Reference
Li, Yang
FerienAkademie 2008
 Jan S Hesthaven, Tim Warburton: Nodal Discontinuous Galerkin
Methods: Algorithms, Analysis, and Applications, Springer
 Cockburn B, Shu CW: TVB Runge-Kutta local projection
discontinuous Galerkin finite element method for conservation laws
II: general framework, MATHEMATICS OF COMPUTATION, v52
(1989), pp.411-435.
 http://lsec.cc.ac.cn/lcfd/DGM_mem.html
 http://www.wikipedia.org/
 http://www.nudg.org/