MCS 355 Scientific Computing

Day 1: Course Introduction

Gustavus Adolphus CollegeSpring 2012

Learning Objectives• Understand the mathematical algorithms used in

scientific computing• Understand error analysis and error propagation in

numerical algorithms• Understand how computational science is used in

modeling scientific applications• Understand the underlying mathematics of calculus and

linear algebra needed for computational science• Develop programming skill at implementing numerical

algorithms • Develop confidence in creating computational solutions

to scientific applications

• What is Scientific Computing?1. Given a scientific or mathematical problem.2. Create a mathematical model.3. Create an algorithm to numerically find a

solution to the model.4. Implement the algorithm in a program.5. Analyze the robustness (accuracy, speed) of the

algorithm. Adjust the algorithm, if needed. 6. Adjust Model, if necessary, go back to 3.

(Feedback Loop)

Let’s Start!!

• Infinite process -> finite process• Non-linear -> linear approximation • Continuous -> discrete • Complex -> simplified

Scientific Computing Reductions

• CAD – Computer-Aided Design• CAM - Computer-Aided Manufacturing• Fluid Flow – Weather models, airplanes• Optimization – business, government, labs • Prototyping – Virtual Models in Car Design• Econometrics – financial models • Signal Processing – Video, Wireless algorithms

Application Areas

• Differential Calculus, Taylor’s Theorem• Integral Calculus• Linear Algebra• Differential Equations

Mathematical Background

• Computer Science I, or some programming experience.

• Matlab is not hard to learn, coding should come fairly easy.

• Will give out lots of example code

Programming Background

Archimedes Principle: The buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.

Example

Archimedes Principle: The buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.

• Exercise: An iron anchor weighs 250 pounds and has a weight density of 480 lbs/ft3. If it totally immersed in sea water that has a weight density of 62.4 lbs/ft3, how much force would be required to lift it while it is immersed?

• Answer: The volume of the water displace by the anchor would be 250/480 (~0.521) cubic feet. Thus, the water will exert a buoyant force of 0.521*62.4 ~ 32.51 lbs. Thus it will take 250-32.51 ~ 217.49 lbs of force to lift the anchor.

Example

Problem: Determine the depth of an object in water without submerging it.

Reductions: finite process, discrete process? probably Non-Linearity – Can’t tell yet. Simplification: Object = sphere of uniform density Density = ρ lbs/ft3

Volume of sphere: 4/3π R3

Simplify: R= 1 Weight = 4/3π ρ

Example

Problem: Determine the depth of an object in water without submerging it.

Matlab: x = linspace(0,1); y = x.^3 – 3*x.^2 + 1; plot(x,y);

Example