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CPS615 Foils -- Master set G for Iterative Approachs to PDE Solution

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

Table of Contents for CPS615 Foils -- Master set G for Iterative Approachs to PDE Solution

001 Iterative Solver Module
    CPS 615 -- Computational Science in
    Simulation Track
    Solution of Simple Partial Differential Equations and Iterative 
    Solvers
    Fall Semester 1995
002 Abstract of PDE and Iterative Solver CPS615 Module
003 Partial Differential Equations: Use in Continuum Physics 
    Examples and basic Notation
004 Examples of Different Types of Partial Differential Equations:
    The Wave Equation (Hyperbolic) and Typical One Dimensional 
    Solution
005 Examples of Different Types of Partial Differential Equations:
    The Parabolic Equation
006 Examples of Different Types of PDE's: Laplace and Poisson Elliptic
     Equations
007 What Conditions are sufficient for solution of PDE's -- Cauchy 
    Boundary Conditions and Hyperbolic,Parabolic and Elliptic PDE's 
008 Closed Boundaries; Dirichlet and Neumann Conditions
    Summary of what PDE Types have What Boundary Conditions
009 Examples of Open(Diffusion) and Closed(Laplace) Boundary 
    Conditions
010 Solutions to Elliptic Equations by Finite Differences
011 Central Difference Operator with error O(h2)
012 Difference Equation form of the Operator to solve Laplace's 
    equation
013 The 12 by 12 Block Tridiagonal Equations Coming from Laplace's 
    Equation on a Tiny 5 by 6 Grid
014 General Form of Sparse Matrix Coming from Laplace's Equation - I
015 General Form of Sparse Matrix Coming from Laplace's Equation in 
    two dimensions - II
016 Iterative Methods and Analogy to Diffusion with an Artificial Time
017 Solution of Artificial Time Equation as a Diffusion System 
    Discretized in Space and Time
018 General 2D Artificial Time Diffusion Equation in Iterative Form
019 Traditional Iterative Methods as Special Cases of Artificial Time 
    Diffusion Formalism
020 Simple Iterative Methods: Jacobi, Gauss-Seidel, SOR
021 Matrix Notation for Iterative Methods
022 General Iteration Matrix Splitting and Preconditioning
023 Explicit Form of General Jacobi Iteration 
    in Matrix and Component Formalism
024 The Special Case of Jacobi Iteration for Laplace's Equation
025 Pseudo Code for the Jacobi Method
026 Formalism for Convergence of Stationary Iterative Methods
027 Eigenvalue Analysis of Iterative Methods
028 Estimation of largest Eigenvalue in One Dimension
029 Eigenvalues and Convergence Rate of Jacobi Iteration
030 Difficult and Easy Eigenfunctions Controlling 
    Convergence of Jacobi Iteration
031 Decoupling of Even and Odd Grid Point Updates in Basic Jacobi 
    Iteration
032 Damping of Eigenfunctions of Short and Long Wavelength
033 Extension of Jacobi Eigenvalue Analysis to two or more Dimensions
034 Direct Solution Method for Ax=b
035 Banded Matrix Computational Complexity
036 Comparison of Computational Complexity between Direct and 
    Iterative Methods
037 Memory Use in Direct and Iterative Methods
038 Over Relaxation (SOR) and Relation to Jacobi and Gauss-Seidel
039 Over Relaxation Eigenvalues and Matrices for Jacobi Iteration
040 Jacobi Iteration Eigenvalues as a function of Over Relaxation 
    Parameter
041 Jacobi Relaxation for Over Relaxation Parameter w =1/2
042 Introduction to Gauss-Seidel Iterative Approach