1. 1 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu MA557/MA578/CS557 Lecture 13

2. 2 Recall Notation

3. 3 Inverse Trace Inequality The following inverse trace inequality will be useful:

4. 4 Orthonormal Basis We can scale each Legendre polynomial so that the integral of their square is unity:

5. 5 Proof of Inverse Trace Inequality First we note that:

6. 6 Proof cont Note that: So F~ is a rank one, symmetric, matrix with non-zero eigenvalue and eigenvector:

7. 7 Proof cont So: By the orthonormality of the scaled Legendre polynomials. QED

8. 8 Extension to a General Interval Now consider the interval [a,b] then a simple scaling argument gives:

9. 9 Useful Paper Check out : http://www.dam.brown.edu/scicomp/publications/Reports/Y2002/BrownSC-2002-19.pdf

10. 10 Back To DG Analysis We are now going to check to see how well DG propagates waves. The following is taken from –S.J.Sherwin, “Dispersion Analysis of the Continuous and Discontinuous Galerkin Formulations”, from “Discontinuous Galerkin Methods: Theory, Computation and Applications”, Ed.s, B.Cockburn, G.Em Karniadakis and C.-W. Shu, Lecture Notes in Computational Science and Engineering, Springer 2000.

11. 11 In Matrix Notation For a 4 cell case with periodicity the matrix system looks like:

12. 12 DG Phase Analysis So let’s seek solutions of the form: For periodicity

13. 13 Plug in.. Here we have a generalized eigenvalue problem for theta since we know D,M,F,G Dimension of the matrix is (p+1)x(p+1)

14. 14 P=0 DG Phase Analysis

15. 15 What does this mean In the j’th cell: The solution looks like: rho(omega*(t-u*x) + dissipation error + phase error)

16. 16 Interpretation 1) The analysis points out that the numerical waves do not travel with speed ubar dispersion 2) The waves decay in time (dissipation) dissipationdispersion

17. 17 Homework 5 1)Find the analytic values of the multiple roots of the matrix for p=0,1,2 with 2)For each p=0,1,2 plot the graphs of each of the solutions 3) Compute the analytic first and second order “error terms” for the phase/dissipation for p=1

18. 18 Homework 5 cont USE Mathematica, Maple or similar to compute the possible multiple analytic eigenvalues Due Friday 02/21/03 DO NOT SPEND 30 HOURS ON THIS. SPEND NO MORE THAN 3 HOURS MAX