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CPS615 Module on Iterative PDE Solvers

Given by Geoffrey C. Fox at CPS615 Basic Simulation Track for Computational Science on Fall Semester 95. Foils prepared 8 November 1995

This Introduces the three fundamental types of PDE's -- Elliptic, Parabolic and Hyperbolic and studies the numerical solution of Elliptic Equations
The sparse matrix formulation is used and iterative approachs -- Jacobi, Gauss Seidel and SOR are defined
These are motivated by analogies between equilibrium of diffusive equations and elliptic systems
Eigenvalue analysis is used to discuss convergence of methods

This mixed presentation uses parts of the following base foilsets which can also be looked at on their own!
CPS615 Foils -- Master set G for Iterative Approachs to PDE Solution
CPS615 Foils on Finite Element Methods, Gauss Seidel, Conjugate Gradient and Differential Operators

Table of Contents for CPS615 Module on Iterative PDE Solvers

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Important references are
1 Iterative Solver Module
CPS 615 -- Computational Science in
Simulation Track
Solution of Simple Partial Differential Equations and Iterative Solvers
Fall Semester 1995
2 Abstract of PDE and Iterative Solver CPS615 Module
Introduction to PDE's and their Classification

3 Partial Differential Equations: Use in Continuum Physics
Examples and basic Notation
4 Examples of Different Types of Partial Differential Equations:
The Wave Equation (Hyperbolic) and Typical One Dimensional Solution
5 Examples of Different Types of Partial Differential Equations:
The Parabolic Equation
6 Examples of Different Types of PDE's: Laplace and Poisson Elliptic Equations
7 What Conditions are sufficient for solution of PDE's -- Cauchy Boundary Conditions and Hyperbolic,Parabolic and Elliptic PDE's
8 Closed Boundaries; Dirichlet and Neumann Conditions
Summary of what PDE Types have What Boundary Conditions
9 Examples of Open(Diffusion) and Closed(Laplace) Boundary Conditions
Discretization of Laplace's equation and Sparse Matrix Form

10 Solutions to Elliptic Equations by Finite Differences
11 Central Difference Operator with error O(h2)
12 Difference Equation form of the Operator to solve Laplace's equation
13 The 12 by 12 Block Tridiagonal Equations Coming from Laplace's Equation on a Tiny 5 by 6 Grid
14 General Form of Sparse Matrix Coming from Laplace's Equation - I
15 General Form of Sparse Matrix Coming from Laplace's Equation in two dimensions - II
Artificial Time Motivation of Iteration

16 Iterative Methods and Analogy to Diffusion with an Artificial Time
17 Solution of Artificial Time Equation as a Diffusion System Discretized in Space and Time
18 General 2D Artificial Time Diffusion Equation in Iterative Form
19 Traditional Iterative Methods as Special Cases of Artificial Time Diffusion Formalism
General Formulation of Iterative Solvers

20 Simple Iterative Methods: Jacobi, Gauss-Seidel, SOR
21 Matrix Notation for Iterative Methods
22 General Iteration Matrix Splitting and Preconditioning
Jacobi Iteration

23 Explicit Form of General Jacobi Iteration
in Matrix and Component Formalism
24 The Special Case of Jacobi Iteration for Laplace's Equation
25 Pseudo Code for the Jacobi Method
Convergence of Iterative Methods
26 Formalism for Convergence of Stationary Iterative Methods
27 Eigenvalue Analysis of Iterative Methods
28 Estimation of largest Eigenvalue in One Dimension
29 Eigenvalues and Convergence Rate of Jacobi Iteration
30 Difficult and Easy Eigenfunctions Controlling
Convergence of Jacobi Iteration
31 Decoupling of Even and Odd Grid Point Updates in Basic Jacobi Iteration
32 Damping of Eigenfunctions of Short and Long Wavelength
33 Extension of Jacobi Eigenvalue Analysis to two or more Dimensions
Comparison of Iterative and Direct Methods

34 Direct Solution Method for Ax=b
35 Banded Matrix Computational Complexity
36 Comparison of Computational Complexity between Direct and Iterative Methods
37 Memory Use in Direct and Iterative Methods
Over Relaxation
38 Over Relaxation (SOR) and Relation to Jacobi and Gauss-Seidel
39 Over Relaxation Eigenvalues and Matrices for Jacobi Iteration
40 Jacobi Iteration Eigenvalues as a function of Over Relaxation Parameter
41 Jacobi Relaxation for Over Relaxation Parameter w =1/2
Gauss Seidel Iterative Methods
42 Introduction to Gauss-Seidel Iterative Approach
43 6:Mathematical and Pseudo Code Form of Gauss Seidel Iteration Method
44 7:Mathematical (Matrix) Form of Gauss Seidel
Parallelism in Gauss Seidel

45 8:Parallelism in Gauss-Seidel Iteration
46 9:Matrix Example Stencil
47 10:Matrix---Wavefront Parallelism for Gauss Seidel
48 11:The Red-Black Two Phase Parallel Gauss Seidel Iteration
49 12:Analysis of Parallel Red Black Gauss Seidel
Convergence of Gauss Seidel

50 13:Eigenvalues of Gauss Seidel Iteration Matrix
51 14:Comparison of Convergence of Gauss-Seidel and Jacobi Iteration
Successive Over Relaxation

52 15:Successive Overrelaxation Iteration Method (SOR)
53 16:Convergence of SOR Compared to Jacobi and Gauss Seidel
54 17:Estimate of Over Relaxation Parameter
55 18:Pseudo Code for SOR---Successive Over Relaxation
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key csepPDE1 URL http://www.npac.syr.edu/projects/csep/pde/pde.html * Basic CSEP PDE Discussion by gcf on Nov 8 95
Times 1 Foils referenced Script
key csepPDE2 URL http://www.npac.syr.edu/projects/csep/pde2/pde2.html * CSEP PDE Discussion of Examples by gcf on Nov 8 95
Times 1 Foils referenced Script
key templatesbook URL http://www.netlib.org/linalg/html_templates/report.html * Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods by gcf on Nov 8 95
Times 1 Foils referenced Script
© on Tue Oct 28 1997