Spring 2006: ATMS 581/AMATH 586/MATH 586

Numerical Analysis of Time Dependent Problems
MWF 2:30-3:20, Room ATG 310C
Final Exam: Tuesday, June 6, 2:30-4:20 PM

Instructor: Professor Dale Durran
606 ATG, 543-7440, durrand@atmos.washington.edu
Office hours: Monday 3:30-4:30,  Thursday 2:00-3:00

Overview: The purpose of the course is to obtain a thorough understanding of the basic numerical techniques that form the foundation for the computer models commonly used to simulate time dependent systems. The theoretical properties of each basic numerical technique will be illustrated by constructing simple models using MATLAB. Some modeling results will be presented as demonstrations in class; other modeling exercises will be assigned as homework.

---

Homework

Exercise 1 (pdf) Download matlab program for Exercise 1. Due Friday April 14th.

Exercise 2 (pdf) Download matlab program for Exercise 2. Due Wednesday April 26th.

• numerical solutions for 4a
• numerical solutions for 4b

Exercise 3 (pdf) This is the take-home midterm. Please work independently. Due Wednesday May 10th.

Exercise 4 (pdf) Download matlab program for Exercise 4. Due Monday May 22nd.

• numerical solutions for 3b
• numerical solutions for 3c

Exercise 5 (pdf) Download matlab program1 and program 2 for Exercise 5. Due Friday June 2nd.

• FFT information sheet (pdf)
  

Take home final (pdf) due Wednesday June 7th by 4 PM.

---

Reading

Introduction

• Two prototypical physical problems
  1. Diffusion dominated
  2. Advection dominated
• Overview of numerical solution strategies.
  1. Finite difference, finite volume
  2. Spectral, finite element

Ordinary Differential Equations

• Single-step, single-stage methods for IVP
  1. Accuracy, consistency, local and global errors
  2. Stability, convergence
• In search of better schemes
  1. Multi-stage methods
     1. Absolute stability region for Runge-Kutta Schemes of orders 1-4
  2. Multi-step methods
  3. Stiff ODEs
• Historical timeline (from the Wikipedia)

Advection Equations

• Upstream differencing
  1. Stability, convergence
  2. The CFL condition
• Higher-order methods
  1. Truncation error and the modified equation
  • Dispersion of a spike (mpeg) (gif)
  • 1st, 2nd, 4th order spike (mpeg) (gif)
  • 1st, 2nd, 4th order sum of 7.5 and 10 Dx waves (mpeg) (gif)
  • 1st, 2nd, 3rd order spike (mpeg) (gif)
  • 1st, 2nd, 3rd order sum of 7.5 and 10 Dx waves (mpeg) (gif)
  2. User-controlled dissipation
     • 3rd, 4th-order explicit and LC compact comparison
• Lax-Wendroff methods