Euler’s Method

CS 170:Computing for the Sciences

and Mathematics

Administrivia

Last time (in P265) Basics of Algorithms and computation

Today More Maple Euler’s method HW #6 Due! HW #7 assigned

Euler’s method

Simplest simulation technique for solving

differential equation

Intuitive

Some other methods faster and more

accurate

Error on order of ∆t

Cut ∆t in half cut error by half

Euler’s Method

Euler’s Method is a simulation technique.

Example: unconstrained growth dP/dt =

0.1P with P0 = 100

P(t) = P(t - ∆t) + growth(t)∆t (new = old + change)

growth(t) is dP/dt = 0.1P(t - ∆t) (change = r*Pold)

Example

dP/dt = 0.1P with P0 = 100 and ∆t = 8

Approximating Unconstrained Growth

initialize simulationLength, population, growthRate, ∆t

numIterations simulationLength / ∆t

for i going from 1 to numIterations do the following:growth growthRate * populationpopulation population + growth * ∆tt t + ∆t

Approximating Unconstrained Growth (Better)

initialize simulationLength, population, growthRate, ∆t

numIterations simulationLength / ∆t

for i going from 1 to numIterations do the following:growth growthRate * populationpopulation population + growth * ∆tt i*∆t

Error

Analytical solution: P = 100 e0.10t

After 100 time units: P = 2,202,647

Smaller ∆t better estimation

∆t = 1 estimate of P = 1,378,061

∆t = 0.25 estimate of P = 1,947,808

∆t = 0.01 estimate of P = 2,191,668

∆t = 0.005 estimate of P = 2,197,149

HOMEWORK!

Homework 7 Maple Tutorial 2 Complete the worksheet’s questions and turn it in Also complete Module 5.2’s Project 1 on page 167. Due 11/1/2010

Thursday’s Class in P115