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Spring 2003Prof. Tim Warburtontimwar@math.unm.edu

MA557/MA578/CS557Lecture 8

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Discontinuous Galerkin Scheme

• Recall that under sufficient conditions (i.e. smoothness) on an interval we obtained the density advection PDE

• We again divide the pipe into sections divided by x1,x2,…,xN .

• On each section we are going to solve a weak version of the advection equation.

0q qut x

1

1 1

1

Find , such that ,i

i

p pi i i i i

xi i

i i i i ix

P x x v P x x

v u dx v x u x xt x

3

1

1 1

1

Find , such that ,i

i

p pi i i i i

xi i

i i i i ix

P x x v P x x

v u dx v x u x xt x

In the i’th cell we will calculate an approximation of q, denoted by . Typically, this will be a p’th order polynomial on the i’th interval. We will solve the following weak equation for i=1,..,N-1

i

x1 x2 x3 xNxN-1

12

31N

i

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Discretization of Pp

• In order to work with the DG scheme we need to choose a basis for Pp and formulate the scheme as a finite dimensional problem

• We expand the density in terms of the Legendre polynomials

1

1 1

1

Find , such that ,i

i

p pi i i i i

xi i

i i i i ix

P x x v P x x

v u dx v x u x xt x

,0

,

1

1

where: is the m'th coefficient of the Legendre

expansion of the density in the i'th cell.2

m p

i i m mm

i m

i i

i i

L x

x x xxx x

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Discretization of Pp

• The scheme:

• Becomes:

1

1 1

1

Find , such that ,i

i

p pi i i i i

xi i

i i i i ix

P x x v P x x

v u dx v x u x xt x

,

1,

0 1

1

, 1, ,0 0 01

Find 0,..., such that

1 1 1 1

i m

m pi m

n mm

m p m p m pm

n i m n i m m n i m mm m m

m p

ddx L L dxdx dt

dLdxu L dx uL L uL Ldx dx

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Simplify, simplify• Starting with:

• We define two matrices:

1

1

1

1

1

1

1

1

nm n m

nm n m

n m

n m

dxM L x L x dxdx

dx dD L x L x dxdx dx

dx dx dL x L x dxdx dx dx

dL x L x dxdx

,

1,

0 1

1

, 1, ,0 0 01

Find 0,..., such that

1 1 1 1

i m

m pi m

n mm

m p m p m pm

n i m n i m m n i m mm m m

m p

ddx L L dxdx dt

dLdxu L dx uL L uL Ldx dx

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Then:

,

1,

0 1

1

, 1, ,0 0 01

Find 0,..., such that

1 1 1 1

i m

m pi m

n mm

m p m p m pm

n i m n i m m n i m mm m m

m p

ddx L L dxdx dt

dLdxu L dx uL L uL Ldx dx

,

,, 1, ,

0 0 0 0

Find 0,..., such that

1 1 1 1

i m

m p m p m p m pi m

nm nm i m n i m m n i m mm m m m

m p

dM u D uL L uL L

dt

Becomes:

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Questions• This raises the following questions how can

we compute Mnm and Dnm

• We will need some of the following results.

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The Legendre Polynomials(used to discretize Pp)

• Let’s introduce the Legendre polynomials.

• The Legendre polynomials are defined as eigenfunctions of the singular Sturm-Liouville problem:

, 0,1,...mL x m

1 1 ( 1) 0mm

d dLx x x m m L xdx dx

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Closed Form Formula for Lm

• If we choose Lm(1) =1 then:

• Examples:

• Warning: this is not a stable way to compute the value of a Legendre polynomial (unstable as m increases)

( / 2)

2

0

2 21 12

l floor ml m l

m ml

m m lL x x

l m

0

0 11

20 12 0

2 2

1

1 21 10 12

2 4 2 21 3 11 10 2 1 22 2

L x

L x x x

xL x x x

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Stable Formula for Computing Lm • We can also use the following recurrence

relation to compute Lm

• Starting with L0=1 L1=x we obtain:

• This is a more stable way to compute the Legendre polynomials at some x.

1 12 1

1 1m m mm mL x xL x L xm m

22 1 0

3 1 1 3 12 2 2

L x xL x L x x

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The Rodriguez Formula• We can also obtain the m’th order Legendre

polynomial using the Rodriguez formula:

21 12 !

m m

m m m

dL x xm dx

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Orthogonality of the Legendre Polynomials

• The Legendre polynomials are orthogonal to each other:

• Since 2m < m+p

11 2 2

11

1 2 2

1

1 11 2 21 11

1 2 2

1

1 11 12 ! 2 !

1 1 1 12 ! 2 !

1 1 1 12 ! 2 !

1 11 1 12 ! 2 !

0

p mp m

p m p p m m

p mp m

p m p m

p mp m

p m p m

m pp mp

p m m p

d dL x L x dx x x dxp dx m dx

d dx x dxp m dx dx

d dx x dxp m dx dx

dx x dxp m dx

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Miscellaneous Relations

m

12

1

1, -1 x 1

L 1 1

1, -1 x 1

21

1 12

22 1

m

m

m

mm

m

L x

m mdL xdx

m mdLdx

L x dxm

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Differentiation• Suppose we have a Legendre expansion for

the density in a cell:

• We can find the coefficients of the Legendre expansion for the derivative of the function as:

where:

,0

m p

i i m mm

L x

(1),

0

m pi

i m mm

d x L xdx

(1), ,

1 odd

2 1j p

i m i jj mj m

m

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Coefficient Differentiation Matrix

(1), ,

11 odd

(1), ,

01

(1)

1

2 2 1

(2 1) if j>m and j+m oddˆ 0 otherwise

2 ˆ

2 ˆor in matrix-vector notation

j p

i m i jj mi ij m

mj

j p

i m mj i jji i

i ii i

mx x

mD

Dx x

x x

ρ Dρ

i.e. if we create a vector of the Legendre coefficients for rho in the I’th cell then we can compute the Legendre coefficients of the derivative of rho in the I’th cell by pre-multiplying the vector of coefficients with the matrix:

1

2 ˆi ix x

D

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The D Matrix• We can use a simple trick to compute D

• Recall:

• We use:

1

1

1

1

nm n m

nm n m

dxM L x L x dxdx

dx dD L x L x dxdx dx

1

1

1

1

1

1

0

simple change of variables

ˆ

2 ˆ2 1

nm n m

n m

n m

j p

nj jmj

nm

dx dD L x L x dxdx dx

dx dx dL x L x dxdx dx dx

dL x L x dxdx

M D

Dn

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The Vandermonde Matrix• It will be convenient for future applications to be able

to transform between data at a set of nodes and the Legendre coefficients of the p’th order polynomial which interpolates the data at these nodes.

• Consider the I’th cell again, but imagine that there are p+1 nodes in that cell then we have:

• Notice that the Vandermonde matrix defined by is square and we can obtain the coefficients by multiplying both sides by the inverse of V

,0

n=0,...,pm p

i n i m m nm

x L x

nm m nV L x

1, ,

0 0

m p n p

i n nm i m i m i nmnm n

x V V x

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Vandermonde• Reiterate:

– that the inverse of the Vandermonde matrix is very useful for transforming nodal data to Legendre coefficients

– that the Vandermonde matrix is very useful for transforming Legendre coefficients to nodal data

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Results So Far• 0<=n,j,m<=p

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1

1

1

1

0 (n )2

2 n=m = 2 2 1

0 (n m)

(2 1) if j>m and j+m oddˆ 0 otherwise

2 ˆ2 1

i inm n m

i i

mj

mnm n

nm

nm m n

x xM L x L x dx m

x xn

mD

dLD L x x dxdx

Dn

V L x

Mass matrix:

Coeff. Differentiation

Stiffness matrix

Vandermonde matrix

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Homework 3Q1) Code up a function (LEGENDRE) which

takes a vector x and m as arguments and computes the m’th order Legendre polynomial at x using the stable recurrence relation.

Q2) Create a function (LEGdiff) which takes p as argument and returns the coefficient differentiation matrix D

Q3) Create a function (LEGmass) which takes p as argument and returns the mass matrix M

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Homework 3 contQ4) Create a function LEGvdm which returns the following matrix:

Q5) Test them with the following matlab routine:

0 i,j pij j iV L x

3) Compute the Chebychev nodes

4) Build the Vandermonde matrix using theChebychev nodes and Legendre polynomials

7) Build the coefficient differentiation matrix

9) Build the Legendre mass matrix

12) Evaluate x^m at the Chebychev nodes

14) Compute the Legendre coefficients

16) Multiply by Dhat (i.e. differentiate in coeff space)

18) Evaluate at the nodes

20) Compute the error in the derivative

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Homework 3 contQ5cont) Explain the output from this test case

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Next Lecture (9)• We will put everything together to build a 1D

DG scheme for advection…