Download presentation

Presentation is loading. Please wait.

Published byArmani Huggins Modified over 3 years ago

1
Maria Ugryumova Direct Solution Techniques in Spectral Methods CASA Seminar, 13 December 2007

2
Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 2/30 3. The matrix diagonalization techniques

3
1. Introduction 3/30 Constant-coefficient Helmholtz equation Some generalizations Spectral descretization methods lead to the system Steady and unsteady problems;

4
Outline 1. Introduction 2. Ad-hoc Direct Methods 3. The matrix diagonalization techniques 4. Direct methods 5. Conclusions 4/30

5
2. Ad-hoc Direct Methods Fourier Chebyshev Legendre 1. To performe appropriate transform 2. To solve the system 3. To performe an inverse transform on to get. Solution process: 5/30 Approximations:

6
2. Ad-hoc Direct Methods Fourier Chebyshev Legendre 1. To performe appropriate transform 2. To solve the system 3. To performe an inverse transform on to get. Solution process: 5/30 Approximations:

7
2.1 Fourier Approximations Problem - the Fourier coefficients; Solution 1a - The Fourier Galerkin approximation The solution is - the trancated Fourier series; 6/30

8
Solution 1b - a Fourier collocation approximation Given 7/30

9
and - the discrete Fourier coefficients; Solution 1b - a Fourier collocation approximation Using the discrete Fourier transform (DFT is a mapping between ) Given 7/30

10
and - the discrete Fourier coefficients; Solution 1b - a Fourier collocation approximation Using the discrete Fourier transform (DFT is a mapping between ) Given (3) is solved for Reversing the DFT 7/30

11
Galerkin and collocation approximation to Helmholz problem are equally straightfoward and demand operations. 8/30

12
2.3 Chebyshev Tau Approximation Solution 1 - Chebyshev Tau approximation: Problem 9/30

13
2.3 Chebyshev Tau Approximation Solution 1 - Chebyshev Tau approximation: Rewriting the second derivative, where L is upper triangular. Solution process requires operations Problem 9/30

14
Solution 2 - To rearrange the equations 2.3.1 More efficient solution procedure For q=2 For q=1 in combination with (5) will lead 10/30

15
After simplification To minimize the round-off errors; quasi-tridiagonal system; not diagonally dominant; Nonhomogeneous BC. For even coefficients: 11/30

16
1.Discrete Chebyshev transform; 2. To solve quasi-tridiagonal system; 3. Inverse Chebyshev transform on to get. 2.4 Mixed Collocation Tau Approximation 12/30 Solution process:

17
2.5 Galerkin Approximation Solution: Legendre Galerkin approx. Problem 13/30

18
2.5 Galerkin Approximation Solution: Legendre Galerkin approx. After integration by parts Problem (full matrices) 13/30

19
The same system but An alternative set of basis functions produces tridiagonal system: Then expension is The right-hand side terms in (5) are related to Legendre coefficients : Two sets of tridiagonal equations; O(N) operations 14/30

20
Transformation between spectral space and physical space: The standard Legendre coefficients of the solution can be found via 15/30

21
2.6 Numerical example for Ad Hoc Methods in 1-D Galerkin method is more accurate than Tau methods Roundoff errors are more for Chebyshev methods, significantly for N>1024 Exact solution is 16/30

22
Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 17/30 3. The matrix diagonalization techniques

23
3.1 Schur Decomposition Collocation approx and Legendre G-NI approxim. lead Problem: Solving (6) by Schur decomposition [Bartels, Stewart, 1972] lower-triangular upper-triangular 18/30

24
Computational cost: Solution process: Reduction and to Schur form Construction of F’ Solution of for U’ Transformation from U’ to U. 19/30

25
3.2 Matrix Digitalization Approach Similar to Schur decomposition. The same solution steps. and are diagonalized Operation cost: 20/30

26
3.3 Numerical example for Ad Hoc Methods in 2-D Problem: Matrix diagonalization was used for the solution procedure Haidvogel and Zang (1979), Shen (1994) Results are very similar to 1-D case 21/30

27
Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 22/30 3. The matrix diagonalization techniques

28
4. Direct Methods Matrix structure produced by Galerkin and G-NI methods ; How the tensor-product nature of the methods can be used efficiently to build matrices; How the sparseness of the matrices in 2D and in 3D can be accounted in direct techniques 23/30

29
4.1 Multidimensional Stiffness and Mass Matrices + homogen. BC onProblem: Integral formulation: Galerkin solution: – stiffness matrix Let be a finite tensor-product basis in. The trial and test function will be chosen in 24/30

30
Decomposition of K into its 1 st, 2 nd, 0 – order components 25/30

31
then Decomposition of K into its 1 st, 2 nd, 0 – order components for a general, the use of G-NI approach with Lagrange nodal basis 26/30 will lead to diagonal marix

32
Decomposition of K into its 1 st, 2 nd, 0 – order components - tensor-product function, - arbitrary, G-NI approach leads to sparse matrix (a matrix-vector multiply requires operations) 27/30 2D3D

33
In 2D: matrix is in general full for arbitrary nonzero Decomposition of K into its 1 st, 2 nd, 0 – order components for arbitrary, G-NI approach (with Lagange nodal basis) leads (a matrix-vector multiply requires operations) In 3D: has sparse structure 28/30 to sparse matrix

34
- decomposition 4. Gaussian Elimination Techniques 29/30 LU - decomposition 2D: special cases of Ad-hoс methods have lower cost Frontal and multifrontal [Davis and Duff 1999] To get benifit from sparsyty, reodering of matrix to factorization have to be done [Gilbert, 1992, Saad, 1996]

35
Outline 1. Introduction 2. Ad-hoc Direct Methods 4. Direct methods 5. Conclusions 30/30 3. The matrix diagonalization techniques

36
5. Conclusions 31/31 Approximation techniques; Galerkin approximations give more accurate results than other methods; Techniques, which can eliminate the cost of solution on prepocessing stage; Sparcity matrices

37
Thank you for attention.

54
Thank you for attention

Similar presentations

OK

1 RA I Sub-Regional Training Seminar on CLIMAT&CLIMAT TEMP Reporting Casablanca, Morocco, 20 – 22 December 2005 Status of observing programmes in RA I.

1 RA I Sub-Regional Training Seminar on CLIMAT&CLIMAT TEMP Reporting Casablanca, Morocco, 20 – 22 December 2005 Status of observing programmes in RA I.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on perimeter and area of plane figures Ppt on old age problems Ppt on retail sales Ppt on micro windmills Ppt on series and parallel circuits physics Ppt on brand marketing specialist Ppt on regional transport office chennai Ppt on tamper resistant seals Ppt on 21st century skills assessment Ppt on cattle farming for class 9