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MCS 355 Scientific Computing

MCS 355 Scientific Computing. Day 1: Course Introduction Gustavus Adolphus College Spring 2012. Learning Objectives. Understand the mathematical algorithms used in scientific computing Understand error analysis and error propagation in numerical algorithms

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MCS 355 Scientific Computing

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  1. MCS 355 Scientific Computing Day 1: Course Introduction Gustavus Adolphus College Spring 2012

  2. 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

  3. Let’s Start!! • What is Scientific Computing? • Given a scientific or mathematical problem. • Create a mathematical model. • Create an algorithm to numerically find a solution to the model. • Implement the algorithm in a program. • Analyze the robustness (accuracy, speed) of the algorithm. Adjust the algorithm, if needed. • Adjust Model, if necessary, go back to 3. (Feedback Loop)

  4. Scientific Computing Reductions • Infinite process -> finite process • Non-linear -> linear approximation • Continuous -> discrete • Complex -> simplified

  5. Application Areas • 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

  6. Mathematical Background • Differential Calculus, Taylor’s Theorem • Integral Calculus • Linear Algebra • Differential Equations

  7. Programming 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

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

  9. 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.

  10. 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 = 4/3πρ

  11. 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);

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