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Spring 2003

Prof. Tim Warburtontimwar@math.unm.edu

MA557/MA578/CS557Lecture 10

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This Lecture

• HW3 answers

• Coding time.

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HW3 Q1

Q1) 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.

A1) We use the following recurrence relationship:

0

1

1 1

1

2 1, 0,1,...

1 1m m m

L x

L x x

m mL x xL x L x m

m m

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HW3 Q1 cont

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HW3 Q1 cont• Let’s plot those Legendre polynomials and see how

they behave over the interval [-1,1]

p=0p=1

p=2

p=3 p=4

p=4

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HW3 Q2

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

A2) The differentiation matrix which maps the Legendre coefficients to the Legendre coefficients of the derivative of the function:

(2 1) if j>m and j+m oddˆ 0 otherwise mj

mD

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HW3 Q2 cont

A2cont) The Matlab code is:

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HW3 Q2 cont

• Let’s test the output for p=5:

(2 1) if j>m and j+m oddˆ 0 otherwise mj

mD

Spot on!.

Note that there are onlyp + (p-2) + (p-4).. Non-zero entries

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HW3 Q2 cont

• The form of Dhat

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HW 3 Q3

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

A3) Definition of the mass matrix:

1 2 n=m

2 2 10 (n m)

i i

nm

x xM n

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HW3 Q3 cont

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HW3 Q4

Q4) Create a function LEGvdm which returns the following matrix:

A4) 0 i,j pij j iV L x

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HW3 Q4 cont

p=0 p=1 p=2

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HW3 Q5

Q5) Test them with the following matlab routine:

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HW3 Q5 cont• Note that x^m is exactly interpolated for m<=p when we use p+1 points

and up to p’th order polynomials. x^(p+1) is only approximately interpolated – and hence the derivative is only approximate.

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Note on Q5

• If we were to repeat this on an interval

• We would need to use:

1,n nx x

1

2 ˆn n

d

dx x x

1FVD V F

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HW3 Q6

Q6) Compute the condition number of the generalized Vandermonde matrix constructed with p=0,…40 and p+1 Chebychev points.

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Asymptotic Behavior cond(V) as p->infinity

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1, ,

0

, ,

% initial condition

for i=N-1:1

for m=0:p

0

end

end

% start time stepping

for tstep=1:Ntsteps

% initiate sigma

for i=N-1:1

for n=0:p

j p

i m i jmjj

i n i n

t V q x

, , , 1,0 01

end

end

% start Runge-Kutta loop

for rk=3:-1:1

for i=N-1:2

for n=0:p

2 1 1 1

end

m p m pn m n

i n i n nm i m i mm mi i

dt nu D u

rk x x

1, 1, 1,02 1

, ,

end

for n=0:p

2 1 1

end

end % end RK loop

for i=N-1:1

for n=0:p

n=0,..,p

en

m pn m

n n nm mm

i n i n

dt nu D

rk x x

1, ,

0

d

end

end

% finish by evaluating solution at cheby nodes

for i=N-1:1

for j=0:p

,

end

end

m p

i j i mjmm

t T x V

Cliff note version of DG advection

equation

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Workshop Code

• Using your own routines or those on the web site code up the DG advection equation solver by Wednesday end of class.

• Spare time this class for coding…