discontinuous galerkin methods

Discontinuous Galerkin Methods

Li, YangFerienAkademie 2008

Li, Yang FerienAkademie 2008

Contents

Working with 1-Dimension

Introduction of DG Methods

Methods of solving PDEs



Finite Difference Method

Finite Volume Method

Finite Element Method

PDEsPDEs



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



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



Simple to implement

Finite Difference

Method

Ill-suited to deal with complex geometries

Element-based discretization

to ensure geometry flexibility



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.



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



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



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.



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



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



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



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



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


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/

discontinuous galerkin methods

Documents

generating element

usual finiteelement

local solution

local residual

space of basis functions

space of test functions

approximate solution

order accurate methods