Chapter 11 polar coordinate

1. Chapter 11 polar coordinate Speaker: Lung-Sheng Chien Book: Lloyd N. Trefethen, Spectral Methods in MATLAB

2. Discretization on unit disk Consider eigen-value problem on unit disk with boundary condition We adopt polar coordinate , then , on and Usually we take periodic Fourier grid in , and non-periodic Chebyshev grid in 1 Chebyshev grid in Chebyshev grid in Observation: nodes are clustered near origin , for time evolution problem, we need smaller time-step to maintain numerical stability. 2 1 to 2 mapping

3. Asymptotic behavior of spectrum of Chebyshev diff. matrix In chapter 10, we have showed that spectrum of Chebyshev differential matrix (second order) approximates with eigenmode 1 Eigenvalue of is negative (real number) and Large eigenmode of does not approximate to 2 Since ppw is too small such that resolution is not enough Mode N is spurious and localized near boundaries

4. Grid distribution 1 2

5. Preliminary: Chebyshev node and diff. matrix [1] Consider Chebyshev node on for Uniform division in arc Even case: Odd case:

6. Preliminary: Chebyshev node and diff. matrix [2] Given Chebyshev nodes and corresponding function value We can construct a unique polynomial of degree , called is a basis. where differential matrix is expressed as for for where with identity Second derivative matrix is

7. Preliminary: Chebyshev node and diff. matrix [3] and Let be the unique polynomial of degree with define and for , then impose B.C. ,that is, We abbreviate In order to keep solvability, we neglect ,that is, zero neglect zero neglect Similarly, we also modify differential matrix as

8. Preliminary: DFT [1] Given a set of data point with is even, Then DFT formula for for for Definition: band-limit interpolant of , is periodic sinc function If we write , then Also derivative is according to

9. Preliminary: DFT [2] , we have Direct computation of derivative of Example: is a Toeplitz matrix. Second derivative is

10. Preliminary: DFT [3] For second derivative operation second diff. matrix is explicitly defined by using Toeplitz matrix (command in MATLAB)

11. Fornberg’s idea : extend radius to negative image [1] and (odd): to avoid singularity of coordinate transformation 1 (even): to keep symmetry condition 2 coordinate coordinate

12. Fornberg’s idea : extend radius to negative image [2] In general is odd, and is even, then and Active variable , total number is coordinate coordinate

13. Redundancy in coordinate transformation [1] 2 to 1 mapping coordinate coordinate redundant

14. Redundancy in coordinate transformation [2] 2 to 1 mapping coordinate coordinate redundant

15. Redundancy in coordinate transformation [3] is expressed as is odd, then Chebyshev differential matrix 8.5 -10.4721 2.8944 -1.5279 1.1056 -0.5 2.6180 -1.1708 -2 0.8944 -0.618 0.2764 -0.7236 2 -0.1708 -1.6180 0.8944 -0.382 0.3820 -0.8944 1.618 0.1708 -2 0.7236 -0.2764 0.6180 -0.8944 2 1.1708 -2.618 0.5 -1.1056 1.5279 -2.8944 10.4721 -8.5 -1.1708 -2 0.8944 -0.618 2 -0.1708 -1.6180 0.8944 -0.8944 1.618 0.1708 -2 0.6180 -0.8944 2 1.1708 neglect

16. Redundancy in coordinate transformation [4] Symmetry property of Chebyshev differential matrix : -1.1708 -2 0.8944 -0.618 -1.1708 -2 is symmetric 2 -0.1708 -1.6180 0.8944 2 -0.1708 -0.8944 1.618 0.1708 -2 -0.618 0.8944 0.6180 -0.8944 2 1.1708 is NOT symmetric 0.8944 -1.6180 Permute column by -1.1708 -2 -0.618 0.8944 2 -0.1708 0.8944 -1.6180 is faster ? -0.8944 1.618 -2 0.1708 0.6180 -0.8944 1.1708 2

17. Redundancy in coordinate transformation [5] is expressed as is odd, then Chebyshev differential matrix 41.6 -68.3607 40.8276 -23.6393 17.5724 -8 21.2859 -31.5331 12.6833 -3.6944 2.2111 -0.9528 -1.8472 7.3167 -10.0669 5.7889 -1.9056 0.7141 0.7141 -1.9056 5.7889 -10.0669 7.3167 -1.8472 -0.9528 2.2111 -3.6944 12.6833 -31.5331 21.2859 -8 17.5724 -23.6393 40.8276 -68.3607 41.6 -31.5331 12.6833 -3.6944 2.2111 7.3167 -10.0669 5.7889 -1.9056 -1.9056 5.7889 -10.0669 7.3167 2.2111 -3.6944 12.6833 -31.5331 neglect

18. Redundancy in coordinate transformation [6] Symmetry property of Chebyshev differential matrix : -31.5331 12.6833 -3.6944 2.2111 7.3167 -10.0669 5.7889 -1.9056 -1.9056 5.7889 -10.0669 7.3167 -31.5331 12.6833 is NOT sym. 2.2111 -3.6944 12.6833 -31.5331 7.3167 -10.0669 Permute column by 2.2111 -3.6944 is NOT sym. -1.9056 5.7889 -31.5331 12.6833 2.2111 -3.6944 7.3167 -10.0669 -1.9056 5.7889 -1.9056 5.7889 7.3167 -10.0669 2.2111 -3.6944 -31.5331 12.6833

19. Row-major indexing: remove redundancy [1] Define active variable for and total number of active variables is 1 NOT 2 3 4 5 6 Index order 7 8 redundant 9 10 11 12 Index order

20. Row-major indexing: remove redundancy [2] is odd, and , then since suppose is even, and Hence for and

21. Row-major indexing: remove redundancy [3] From symmetry condition, we have for and , symmetry condition implies Therefore, we have two important relationships 1 2

22. Kronecker product [1] 1 2 Define active variable 3 4 for and 5 6 Separation of variable: assume matrix A acts on r-dir and matrix B acts on is independent of Let be row-major index of active variable is independent of

23. Kronecker product [2] Kronecker product is defined by Case 1: Case 2:

24. Kronecker product [3] Case 3: permutation, if permute

25. Kronecker product [4] Case 4:

26. Non-active variable  active variable [1] , on and is even, then is odd, and Active variable is , that is Total number is NOT Note that differential matrix is of dimension ,acts on Neglect due to B.C.

27. Non-active variable  active variable [2] However and act on NOT active variable, how to deal with? From previous discussion, we have following relationships which can solve this problem and where is a permutation matrix

28. Non-active variable  active variable [3] Recall for and Consider Chebyshev differential matrix acts on and evaluate at We write in matrix notation Question: How about if we arrange equations on when fixed

29. Non-active variable  active variable [4] abbreviate We only keep operations on active variable That is, only consider equation for Later on, we use the same symbol

30. Non-active variable  active variable [5] Define permutation matrix Let Active variable with the same indexing in r-direction

31. Non-active variable  active variable [6] Moreover, we modify differential matrix according to permutation P by where such that for and Evaluated at

32. Non-active variable  active variable [7] define and active variable To sum up

33. Non-active variable  active variable [8] Note that under row-major indexing, memory storage of is but If we adopt Kronecker products, then Definition: Kronecker product is defined by

34. Non-active variable  active variable [9] The same reason holds for second derivative operator neglect equation on then collect equations on each we have where

35. Non-active variable  active variable [10] We write second derivative operator on as where for so implies

36. Non-active variable  active variable [11] If we adopt Kronecker products, then

37. Non-active variable  active variable [12] summary 1 2 3 Note that so that all three system of equations are of the same order.

38. Non-active variable  active variable [13] , on and Discretization on then where

39. Example: program 28 with boundary condition is even, let eigen-pair be is odd and sparse structure of Dimension :

40. Example: program 28 (mash plot of eigenvector) 1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue Eigenvector is normalized by supremum norm, 2

41. Example: program 28 (nodal set)

42. Exercise 1 on annulus with boundary condition is even, let eigen-pair be is odd and Chebyshev node on Chebyshev node on

43. Exercise 1 (mash plot of eigenvector) 1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue Eigenvector is normalized by supremum norm, 2