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Discontinuous Galerkin Methods. Li, Yang FerienAkademie 2008. Contents. Methods of solving PDEs. Introduction of DG Methods. Working with 1-Dimension. Methods of solving PDEs. Finite Difference Method. PDEs. Finite Volume Method. Finite Element Method. Methods of solving PDEs. - PowerPoint PPT Presentation
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Discontinuous Galerkin Methods
Li, YangFerienAkademie 2008
Li, Yang FerienAkademie 2008
Contents
Working with 1-Dimension
Introduction of DG Methods
Methods of solving PDEs
Li, Yang FerienAkademie 2008
Methods of solving PDEs
Finite Difference Method
Finite Volume Method
Finite Element Method
PDEsPDEs
Li, Yang FerienAkademie 2008
Methods of solving PDEs
E.g. 1D scalar conservation law
with initial conditions and boundary conditions on the boundary
unknown solution
flux
prescribed force
,u f
g xt x
( , )u x t
( )f u
( , )g x t
How to get the approximate solution
?Is it satisfied the
equation?
( , )hu x t
Li, Yang FerienAkademie 2008
Methods of solving PDEs
Finite Difference Method
a grid
local grid size
assume:
Residual:
1 1
1
( , ) ( , ) ( , )( , )
k k kkh h h
k k
du x t f x t f x tg x t
dt h h
, 1...k
px k N1k k kh x x
2 21 1
0 0
[ , ] : ( , ) ( )( ) , ( , ) ( )( )k k k i k ih i h i
i i
x x x u x t a t x x f x t b t x x
( , ) ( , )h hh
u fR x t g x t
t x
Li, Yang FerienAkademie 2008
Methods of solving PDEs
Simple to implement
Finite Difference
Method
Ill-suited to deal with complex geometries
Element-based discretization
to ensure geometry flexibility
Li, Yang FerienAkademie 2008
Methods of solving PDEs
Finite Volume Method
element staggered grid ,
solution is approximated on the element by a constant
( ): ( , ) ( , )
k k
kh
u f ux D R x t g x t
t x
1/ 2 1/ 2[ , ]k k kD x x
( )ku t
1/ 2 11( )2
k k kx x x
1/ 2 1/ 2
kkk k k kdu
h f f h gdt
Divergence Theorem
Reconstruction of Solution hu 0
( ) ( )p
k ih i
i
u x a x x
To find the p+1 unknown coefficients need information at least from p+1 cells
The actual numerical scheme will depend upon problem geometry
and mesh construction.
Difficult when high-order
reconstruction.
Li, Yang FerienAkademie 2008
Methods of solving PDEs
Finite Element Method
assume the local solution:
element locally defined basis function
global representation of :
where is the basis function.
define a space of test functions, , and require the residual is orthogonal to all test functions:
1
: ( ) ( )Np
kh n n
n
x D u x b x
1[ , ]k k kD x x ( )n x
hu
1
( ) ( ) ( )K
k kh
k
u x u x N x
hV
( ) ( ) 0,h hh h h h
u fg x dx V
t x
( )ij ijN x
Li, Yang FerienAkademie 2008
Methods of solving PDEs
Finite Element MethodClassical choice: the spaces spanned by the basis functions and test functions are the same.
Since the residual has to vanish for all
1
( ) ( ) ( )K
k kh
k
x v x N x
h hV
( ) ( ) 0,jh hh
u fg N x dx
t x
hh h
d
dt
uM Sf Mg
( ) ( ) , ( )j
i j iij ij
dNN x N x dx N x dx
dx M S
Easy to extend to high-order approximation by
adding additional degrees of freedom to
the element.
The semi-discrete scheme becomes
implicit and M must be inverted
Li, Yang FerienAkademie 2008
Introduction of DG Methods
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.
Li, Yang FerienAkademie 2008
Introduction of DG Methods
From FEM and FVM to DG-FEM maintain the definition of elements as in the FEM
but new definition of vector of unknowns
Assume the local solution in each element is: (likewise for the flux)
Define The space of basis functions:
The local residual is:
1 2 2 3 1 1[ , , , , , , , , ]K K K K Th u u u u u u u u u
1[ , ]k k kD x x
1 11
1 10
: ( ) ( )k k
k k k k k i kh ik k k k
i
x x x xx D u x u u u l x
x x x x
11 0
K kh k i iV l
: ( , ) ( , ),k k
k h hh
u fx D R x t g x t
t x
Li, Yang FerienAkademie 2008
Introduction of DG Methods
Require that the residual is orthogonal to all test functions :
Similar to FVM, use Gauss’ theorem:
introduce the numerical flux, , as the unique value to be used at the interface and obtained by coming information from both elements.
applying Gauss’ theorem again:
1
[ ]k
kk
kkjk k k k k xh
j h j h j xD
jul f gl dx f l
t x
h hV
( , ) ( ) 0k
kh iDR x t l x dx
f
1
[ ]k
kk
kkjk k k k xh
j h j j xD
jul f gl dx f l
t x
1
( , ) ( ) [( ) ]k
kk
k k k xh j h j xDR x t l x dx f f l
Weak Form
Strong Form
Li, Yang FerienAkademie 2008
Introduction of DG Methods
More general formConsider the nonlinear, scalar, conservation law:
subject to appropriate initial conditions
The boundary conditions are provided when the boundary is an inflow boundary:
when
when
We still assume that the global solution can be well approximated by a space of piecewise polynomial functions, defined on the union of , and require the residual to be orthogonal to space of the test functions,
( )0, [ , ]
u f ux L R
t x
0( ,0) ( ).u x u x
1( , ) ( )u L t g t ( ( , )) 0,hf u L t
2( , ) ( )u R t g t ( ( , )) 0,hf u R t
kD1
K kh k h hV
Li, Yang FerienAkademie 2008
Introduction of DG Methods
recover the locally defined weak formulation:
and the strong form:
Assume that all local test functions can be represented by using a local polynomial basis, , as
and leads to equations as:
ˆ( ( ) ) ,k k
k kk k k kh hh h h hD D
u df u dx f dx
t dx
n
( )ˆ( ) ( ( ) ) ,
k k
k k kk k k kh h hh h h hD D
u f udx f u f dx
t x
n
( )n x
1
ˆ: ( ) ( )pN
k k kh n n
n
x D x x
pN
ˆ( ( ) ) ,k k
kk kh n
n h h nD D
u df u dx f dx
t dx
n( )
ˆ( ) ( ( ) ) ,k k
k k kk kh h h
n h h nD D
u f udx f u f dx
t x
n
Li, Yang FerienAkademie 2008
Working with 1-Dimension
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
Li, Yang FerienAkademie 2008
Working with 1-Dimension
Li, Yang FerienAkademie 2008
Working with 1-Dimension
Li, Yang FerienAkademie 2008
Reference
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/
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