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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
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This Introduces the three fundamental types of PDE's -- Elliptic, Parabolic and Hyperbolic and studies the numerical solution of Elliptic Equations
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The sparse matrix formulation is used and iterative approachs -- Jacobi, Gauss Seidel and SOR are defined
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These are motivated by analogies between equilibrium of diffusive equations and elliptic systems
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Eigenvalue analysis is used to discuss convergence of methods
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This mixed presentation uses parts of the following base foilsets which can also
be looked at on their own!
CPS615-95G CPS615 Foils -- Master set G for Iterative
Approachs to PDE Solution
CPS615FEMetc95 CPS615 Foils on Finite Element Methods, Gauss
Seidel, Conjugate Gradient and Differential
Operators
Important references are
CPS615-95G 001 001 Iterative Solver Module
CPS 615 -- Computational Science in
Simulation Track
Solution of Simple Partial
Differential Equations and Iterative
Solvers
Fall Semester 1995
CPS615-95G 002 002 Abstract of PDE and Iterative Solver
CPS615 Module
Introduction to PDE's and their Classification
CPS615-95G 003 003 Partial Differential Equations: Use
in Continuum Physics
Examples and basic Notation
CPS615-95G 004 004 Examples of Different Types of
Partial Differential Equations:
The Wave Equation (Hyperbolic) and
Typical One Dimensional Solution
CPS615-95G 005 005 Examples of Different Types of
Partial Differential Equations:
The Parabolic Equation
CPS615-95G 006 006 Examples of Different Types of
PDE's: Laplace and Poisson Elliptic
Equations
CPS615-95G 007 007 What Conditions are sufficient for
solution of PDE's -- Cauchy Boundary
Conditions and Hyperbolic,Parabolic
and Elliptic PDE's
CPS615-95G 008 008 Closed Boundaries; Dirichlet and
Neumann Conditions
Summary of what PDE Types have What
Boundary Conditions
CPS615-95G 009 009 Examples of Open(Diffusion) and
Closed(Laplace) Boundary Conditions
Discretization of Laplace's equation and Sparse Matrix Form
CPS615-95G 010 010 Solutions to Elliptic Equations by
Finite Differences
CPS615-95G 011 011 Central Difference Operator with
error O(h2)
CPS615-95G 012 012 Difference Equation form of the
Operator to solve Laplace's equation
CPS615-95G 013 013 The 12 by 12 Block Tridiagonal
Equations Coming from Laplace's
Equation on a Tiny 5 by 6 Grid
CPS615-95G 014 014 General Form of Sparse Matrix Coming
from Laplace's Equation - I
CPS615-95G 015 015 General Form of Sparse Matrix Coming
from Laplace's Equation in two
dimensions - II
Artificial Time Motivation of Iteration
CPS615-95G 016 016 Iterative Methods and Analogy to
Diffusion with an Artificial Time
CPS615-95G 017 017 Solution of Artificial Time Equation
as a Diffusion System Discretized
in Space and Time
CPS615-95G 018 018 General 2D Artificial Time Diffusion
Equation in Iterative Form
CPS615-95G 019 019 Traditional Iterative Methods as
Special Cases of Artificial Time
Diffusion Formalism
General Formulation of Iterative Solvers
CPS615-95G 020 020 Simple Iterative Methods: Jacobi,
Gauss-Seidel, SOR
CPS615-95G 021 021 Matrix Notation for Iterative
Methods
CPS615-95G 022 022 General Iteration Matrix Splitting
and Preconditioning
Jacobi Iteration
CPS615-95G 023 023 Explicit Form of General Jacobi
Iteration
in Matrix and Component Formalism
CPS615-95G 024 024 The Special Case of Jacobi Iteration
for Laplace's Equation
CPS615-95G 025 025 Pseudo Code for the Jacobi Method
Convergence of Iterative Methods
- Illustrated with Jacobi Case
CPS615-95G 026 026 Formalism for Convergence of
Stationary Iterative Methods
CPS615-95G 027 027 Eigenvalue Analysis of Iterative
Methods
CPS615-95G 028 028 Estimation of largest Eigenvalue in
One Dimension
CPS615-95G 029 029 Eigenvalues and Convergence Rate of
Jacobi Iteration
CPS615-95G 030 030 Difficult and Easy Eigenfunctions
Controlling
Convergence of Jacobi Iteration
CPS615-95G 031 031 Decoupling of Even and Odd Grid
Point Updates in Basic Jacobi
Iteration
CPS615-95G 032 032 Damping of Eigenfunctions of Short
and Long Wavelength
CPS615-95G 033 033 Extension of Jacobi Eigenvalue
Analysis to two or more Dimensions
Comparison of Iterative and Direct Methods
CPS615-95G 034 034 Direct Solution Method for Ax=b
CPS615-95G 035 035 Banded Matrix Computational
Complexity
CPS615-95G 036 036 Comparison of Computational
Complexity between Direct and
Iterative Methods
CPS615-95G 037 037 Memory Use in Direct and Iterative
Methods
Over Relaxation
- Introduction and fruitless Jacobi Example
CPS615-95G 038 038 Over Relaxation (SOR) and Relation
to Jacobi and Gauss-Seidel
CPS615-95G 039 039 Over Relaxation Eigenvalues and
Matrices for Jacobi Iteration
CPS615-95G 040 040 Jacobi Iteration Eigenvalues as a
function of Over Relaxation
Parameter
CPS615-95G 041 041 Jacobi Relaxation for Over
Relaxation Parameter w =1/2
Gauss Seidel Iterative Methods
- Introduction, Matrix Form
CPS615-95G 042 042 Introduction to Gauss-Seidel
Iterative Approach
CPS615FEMetc95 006 043 6:Mathematical and Pseudo Code Form
of Gauss Seidel Iteration Method
CPS615FEMetc95 007 044 7:Mathematical (Matrix) Form of
Gauss Seidel
Parallelism in Gauss Seidel
CPS615FEMetc95 008 045 8:Parallelism in Gauss-Seidel
Iteration
CPS615FEMetc95 009 046 9:Matrix Example Stencil
CPS615FEMetc95 010 047 10:Matrix---Wavefront Parallelism
for Gauss Seidel
CPS615FEMetc95 011 048 11:The Red-Black Two Phase Parallel
Gauss Seidel Iteration
CPS615FEMetc95 012 049 12:Analysis of Parallel Red Black
Gauss Seidel
Convergence of Gauss Seidel
CPS615FEMetc95 013 050 13:Eigenvalues of Gauss Seidel
Iteration Matrix
CPS615FEMetc95 014 051 14:Comparison of Convergence of
Gauss-Seidel and Jacobi Iteration
Successive Over Relaxation
CPS615FEMetc95 015 052 15:Successive Overrelaxation
Iteration Method (SOR)
CPS615FEMetc95 016 053 16:Convergence of SOR Compared to
Jacobi and Gauss Seidel
CPS615FEMetc95 017 054 17:Estimate of Over Relaxation
Parameter
CPS615FEMetc95 018 055 18:Pseudo Code for SOR---Successive
Over Relaxation
List of Foils Used as they occur
CPS615-95G CPS615 Foils -- Master set G for Iterative
Approachs to PDE Solution
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
CPS615FEMetc95 CPS615 Foils on Finite Element Methods, Gauss
Seidel, Conjugate Gradient and Differential
Operators
6 7 8 9 10 11 12 13 14 15 16 17 18
Sorted List of Foils Used
CPS615-95G CPS615 Foils -- Master set G for Iterative
Approachs to PDE Solution
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
CPS615FEMetc95 CPS615 Foils on Finite Element Methods, Gauss
Seidel, Conjugate Gradient and Differential
Operators
6 7 8 9 10 11 12 13 14 15 16 17 18
© on Tue Oct 28 1997