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MA557/MA578/CS557 Lecture 10

MA557/MA578/CS557 Lecture 10. Spring 2003 Prof. Tim Warburton timwar@math.unm.edu. This Lecture. HW3 answers Coding time. HW3 Q1.

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MA557/MA578/CS557 Lecture 10

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  1. MA557/MA578/CS557Lecture 10 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu

  2. This Lecture • HW3 answers • Coding time.

  3. 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:

  4. HW3 Q1 cont

  5. HW3 Q1 cont • Let’s plot those Legendre polynomials and see how they behave over the interval [-1,1] p=0 p=1 p=3 p=4 p=2 p=4

  6. 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:

  7. HW3 Q2 cont A2cont) The Matlab code is:

  8. HW3 Q2 cont • Let’s test the output for p=5: Spot on!. Note that there are onlyp + (p-2) + (p-4).. Non-zero entries

  9. HW3 Q2 cont • The form of Dhat

  10. 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:

  11. HW3 Q3 cont

  12. HW3 Q4 Q4) Create a function LEGvdm which returns the following matrix: A4)

  13. HW3 Q4 cont p=0 p=1 p=2

  14. HW3 Q5 Q5) Test them with the following matlab routine:

  15. 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.

  16. Note on Q5 • If we were to repeat this on an interval • We would need to use:

  17. HW3 Q6 Q6) Compute the condition number of the generalized Vandermonde matrix constructed with p=0,…40 and p+1 Chebychev points.

  18. Asymptotic Behavior cond(V) as p->infinity

  19. Cliff note version of DG advection equation

  20. 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…

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