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CPS615 Foils -- set E: ODE's and Particle Dynamics

Given by Nancy McCracken and Geoffrey C. Fox at CPS615 Basic Simulation Track for Computational Science on Fall Semester 95/96/97. Foils prepared 14 October 1997

This uses the simple O(N2) Particle Dynamics Problem as a motivator to discuss solution of ordinary differential equations
We discuss Euler, Runge Kutta and predictor corrector methods
The simple data parallel O(N2) algorithm is given in Fortran90 and HPF
The better Pipeline version is also given
We analyse Performance

Table of Contents for CPS615 Foils -- set E: ODE's and Particle Dynamics

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1 CPS 615 -- Computational Science in
Simulation Track
Data Parallel Module on ODE's and Particle Dynamics
October 23, 1995
Updated Oct 11,1996
2 Abstract of Data Parallel ODE and Particle Dynamics Module
3 Particle (N-Body) Applications and Ordinary Differential Equations (ODE's)
4 Particle Applications - Ordinary Differential Equations (ODE's)
5 Particle Applications - the N-body problem
6 Newton's First Law -- The Gravitational Force on a Particle
7 Equations of Motion -- Newton's Second Law
8 Numerical techniques for solving ODE's
9 Second and Higher Order Equations
10 Basic Discretization of Single First Order Equation
11 Errors in numerical approximations
12 Runge-Kutta Methods: Euler's method
13 Estimate of Error in Euler's method
14 Relationship of Error to Computation
15 Example using Euler's method from the CSEP book
16 Approximate solutions at t=1,using Euler's method with different values of h
17 Runge-Kutta Methods: Modified Euler's method
18 Approximate solutions of the ODE for et at t=1, using modified Euler's method with different values of h
19 The Classical Runge-Kutta -- In Words
20 The Classical Runge-Kutta -- Formally
21 The Classical Runge-Kutta Pictorially
22 Predictor / Corrector Methods
23 Definition of Multi-step methods
24 Features of Multi-Step Methods
25 Comparison of Explicit and Implicit Methods
26 Solving the N-body equations of motion
27 Representing the Data Parallel N-Body problem
28 Form of the Computation -- Data v. Message Parallel
29 N-body Runge Kutta Routine in Fortran90 - I
30 Runge Kutta Routine in Fortran90 - II
31 Computation of accelerations - a simple parallel array algorithm
32 Simple Data Parallel Version of N Body Force Computation -- Grav -- I
33 The Grav Function in Data Parallel Algorithm - II
34 Some Inefficiencies of the Data Parallel N2 Algorithm - I
35 Some Inefficiencies of the Data Parallel N2 Algorithm - II
36 Better Data Parallel Pipeline Algorithm for Computation of Accelerations,
taking 1/2 the time for iterations over force computation
37 Data Parallel Pipeline Algorithm in detail
38 Basic Data Parallel pipeline operation
39 Examples of Data Parallel Pipeline Algorithm
40 Data Parallel Pipeline Algorithm Grav -- Part I
41 Data Parallel Pipeline Algorithm for Grav -- Part II
42 Data Parallel Grav Pipeline Algorithm, concluded
43 Data Parallel Parallel Decomposition
44 Data Parallel Parallel Execution Time -I
45 Data Parallel Parallel Execution Time -II
46 N-body Problem is a one dimensional Algorithm
47 Excerpts from an HPF program for this algorithm
48 HPF program excerpts - II
49 HPF program excerpts - finished
50 Notes and References
51 CPS 615 -- Computational Science in
Simulation Track
Message Parallel Module on ODE's and Particle Dynamics
October 20, 1997
52 Abstract of Message Parallel ODE and Particle Dynamics
53 Summary of Parallel N-Body Programming Methods and Algorithms
54 Status of Parallelism in Various N Body Cases
55 Other N-Body Like Problems - I
56 Other N-Body Like Problems - II
57 Essential Structure of Message Parallel O(N2) Algorithm - I
58 Essential Structure of Message Parallel O(N2) Algorithm - II
59 Structure of Runge Kutta Phases
60 The 9 Fortran Phases in Runge Kutta Update
61 Features of Message Parallel Computation
62 Message Parallel Force Computation MPGrav
63 Very Bad Naive Message Parallel Algorithm
64 Features of Very Bad Naive Message Parallel Algorithm
65 Much Better Message Parallel Algorithm
66 Blocking of Messages
67 Compilers, Caches and Data Locality
68 Communication and Computation Complexity
69 Best Message Parallel N Body Algorithm - I
70 Best Message Parallel N Body Algorithm - II
71 Choice of Place to Compute Fi,j
72 Final Remarks on Best Algorithm
73 CPS 615 -- Computational Science in
Simulation Track
Data Parallel Module on ODE's and Particle Dynamics
February 20, 1998
74 Abstract of Message Parallel ODE and Particle Dynamics
75 Representing the Message Parallel N-Body problem
76 Form of the Computation -- Message Parallel
77 Pipeline Algorithm in detail
78 Basic Message Parallel pipeline operation
79 Examples of Message Parallel Pipeline Algorithm
80 Factor of Two in the Parallel O(N2) Algorithm
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