Ordinary Differential Equations
Higher Order Schemes
CS370 Lecture 12 May 30, 2016
1
Trapezoidal Rule - Illustrated
+1
= + , + +1 , +1
2
+1
(+1 )
+1
Evaluate the slope
= at the start
and end of the time
step, and step along
the average slope.
( )
+1
2

Interpolation
More Flexible Curves
CS 370 - Lecture 8
May 18, 2016
1
Cubic splines via Hermite approach
Recall: we showed how to use Hermite interpolation ideas to solve for
a cubic spline.
1. Set up a linear system to solve for slopes , by requiring mat

Ordinary Differential Equations
Truncation Error, and
Better Schemes
CS370 Lecture 11 May 27, 2016
1
Last Time ODEs and Forward Euler
Considered 1st order ordinary differential equations (ODE) in the form
= (, ), with 0 = 0 .
We applied forward Euler to

Ordinary Differential
Equations More Schemes
CS370 Lecture 13 June 1, 2016
1
Time Integration Methods So Far
Forward Euler: Use slope at starting point.
Local Truncation Error (LTE): 2 . Explicit.
Trapezoidal: Use average of slope at start and end of ste

Introduction to Interpolation
CS370 May 9, 2016
1
Round v.s. Truncation - Clarification
Floating point systems offer different rounding modes.
We considered:
1) Round-to-nearest () rounds to closest available number in .
Usually the default.
Well break

Ordinary Differential Equations
Determining Truncation Error
CS370 Lecture 16 June 8, 2016
1
Administrative Stuff
The midterm is on Thursday June 16, at 7pm-8:50pm in M3 1006.
I have posted a list of topics and associated course note section
numbers on P

Ordinary Differential
Equations Introduction
CS370 Lecture 10 May 25, 2016
1
Today
What is an ODE (ordinary differential equation)?
What form do they have?
Why do we need numerical methods to solve them?
What is the simplest such method?
2
Differentia

Ordinary Differential Equations
Converting to First Order Systems
CS370 Lecture 14 June 3, 2016
1
Application: Predator-Prey Population
Dynamics (Lotka Volterra equation)
=
= ()( )
() is prey population, () is predator population.
2
High Order ODEs
In

Interpolation
Efficiently Computing Splines
CS370 Lecture 7
May 16, 2016
1
Two strategies for smooth curves
Hermite interpolation: Given points
and their slopes, fit a curve. Gives
matching 1st derivatives between
intervals.
Cubic spline interpolation: G