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CPS615-Basic PDE Solver Discussion and Sparse Matrix Formulation

Given by Geoffrey C. Fox at Delivered Lectures of CPS615 Basic Simulation Track for Computational Science on 8 November 96. Foils prepared 11 November 1996

This starts basic module on Partial Differential Solvers with
Introduction to Classification of Equations
Basic Discretization
Derivation of Sparse Matrix Formulation
Analogies of Iteration with Artificial Time
Start of Explicit Matrix Formulation for Simple Cases

This mixed presentation uses parts of the following base foilsets which can also be looked at on their own!
Master Set of Foils for 1996 Session of CPS615
CPS615 Foils -- Master set G for Iterative Approachs to PDE Solution

Table of Contents for CPS615-Basic PDE Solver Discussion and Sparse Matrix Formulation

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CPS 615 Lectures 1996 Fall Semester -- November 8
Introduction to PDE's
1 Delivered Lectures for CPS615 -- Base Course for the Simulation Track of Computational Science
Fall Semester 1996 --
Lecture of November 8 - 1996
2 Abstract of Nov 8 1996 CPS615 Lecture
3 Abstract of PDE and Iterative Solver CPS615 Module
Introduction to PDE's and their Classification

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

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

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

21 Simple Iterative Methods: Jacobi, Gauss-Seidel, SOR
22 Matrix Notation for Iterative Methods
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