Numerical Methods for Partial Differential Equations: an Overview and Applications André Jaun, NADA, Royal Institute of Technology, Stockholm Register now for the Netuniversity edition (10 Oct - 16 Dec 2005)

This is the web edition of the distance learning courses taught at the Royal Institute of Technology in Stockholm (KTH course 2D5246, 4 points), the Swedish Netuniversity (KTH course 2D4232, 4 points), other universities and independent learners from outside Sweden. The target in 4 weeks time, is to provide an introduction in computational methods for graduate- and lifelong learning students, using a flexible and applied learning method that can easily be tailored to professional schedules. Short video recordings introduce the subject by following the teacher's line of thought; the material is then studied and assimilated in a problem based learning environement, performing numerical experiments in the JBONE applet below: be patient, it takes a one time only 70 secs to load with a modem...

JBONE applet  press START/STOP to simulate the propagation of a shock wave computed with a finite difference approximation of the Burger equation. You may re-scale the plot area using Toggle display and edit the parameters with a click of the mouse button.

Numerical methods such as finite differences, finite elements, fast Fourier transforms, Monte-Carlo and Lagrangian schemes are discussed for a variety of problems including the advection, diffusion, Black-Scholes, Burger, Korteweg-DeVries and the Schroedinger equation. Assignments are submitted from the web browser and are automatically compiled into web pages where the students explain with their own words, equations and programs how to derive, implement and run their own numerical schemes.

Recommended by

You can help us and sponsor this website to keep it accessible free of charge for everyone using a simple click below:

	US$ 10 (e-paperback), US$ 35 (e-book), US$ 100 (student). Thank you very much and enjoy...

Table of contents

• Contents
• 1 INTRODUCTION
  • 1.1 About this course
    • 1.1.1 Motivation
    • 1.1.2 How to study
  • 1.2 Differential Equations
    • 1.2.1 Ordinary differential equations
    • 1.2.2 Partial differential equations
    • 1.2.3 Boundary / initial conditions
    • 1.2.4 Characteristics and dispersion relations
    • 1.2.5 Moments and conservation laws
  • 1.3 Prototype problems
    • 1.3.1 Advection
    • 1.3.2 Diffusion
    • 1.3.3 Dispersion
    • 1.3.4 Wave-breaking
    • 1.3.5 Schrödinger
  • 1.4 Numerical discretization
    • 1.4.1 Convergence, Richardson extrapolation
    • 1.4.2 Sampling on a regular grid
    • 1.4.3 Finite elements
    • 1.4.4 Splines
    • 1.4.5 Harmonic functions
    • 1.4.6 Wavelets
    • 1.4.7 Sampling with quasi-particles
  • 1.5 Computer quiz
  • 1.6 Exercises
  • 1.7 Further reading
  • 1.8 Interactive evaluation form
  
  
• 2 FINITE DIFFERENCES
  • 2.1 Explicit 2 levels
  • 2.2 Explicit 3 levels
  • 2.3 Lax-Wendroff
  • 2.4 Leap-frog, staggered grids
  • 2.5 Implicit Crank-Nicholson
    • 2.5.1 Advection-diffusion equation
    • 2.5.2 Schrödinger equation
  • 2.6 Computer quiz
  • 2.7 Exercises
  • 2.8 Further reading
  • 2.9 Interactive evaluation form
  
  
• 3 FINITE ELEMENT METHOD
  • 3.1 Mathematical background
  • 3.2 An engineer's formulation
  • 3.3 Numerical quadrature
  • 3.4 Implementation and solution
  • 3.5 Linear solvers
  • 3.6 Variational inequalities
  • 3.7 Computer quiz
  • 3.8 Exercises
  • 3.9 Further Reading
  • 3.10 Interactive evaluation form
  
  
• 4 FOURIER TRANSFORM
  • 4.1 FFT with the computer
  • 4.2 Linear equations.
  • 4.3 Aliasing, filters and convolution.
  • 4.4 Non-linear equations.
  • 4.5 Computer quiz
  • 4.6 Exercises
  • 4.7 Further Reading
  • 4.8 Interactive evaluation form
  
  
• 5 MONTE-CARLO METHOD
  • 5.1 Monte Carlo integration
  • 5.2 Stochastic theory
  • 5.3 Particle orbits
  • 5.4 A scheme for the advection diffusion equation
  • 5.5 When should you use Monte Carlo methods?
  • 5.6 Computer quiz
  • 5.7 Exercises
  • 5.8 Further Reading
  • 5.9 Interactive evaluation form
  
  
• 6 LAGRANGIAN METHOD
  • 6.1 Splitting advection from the rest
  • 6.2 Cubic-Interpolated Propagation (CIP)
  • 6.3 Non-Linear equations with CIP
  • 6.4 Quiz
  • 6.5 Exercises
  • 6.6 Further Reading
  • 6.7 Interactive evaluation form
  
  
• 7 WAVELETS
  • 7.1 Remain a matter of research
  
  
• 8 THE JBONE USER MANUAL
  • 8.1 Source code & installation
  • 8.2 Program execution
  • 8.3 Program structure & documentation
  • 8.4 An object-oriented example: Monte Carlo in JBONE
  • 8.5 Interactive evaluation form
  
  
• 9 PROBLEM BASED LEARNING ENVIRONEMENT
  • 9.1 Typesetting with TEX
  • 9.2 Programming in JAVA
  • 9.3 Parameters and switches in HTML
  • 9.4 Interactive evaluation form
  
  
• 10 COURSE EVALUATION AND PROJECTS
  • 10.1 Interactive evaluation form
  • 10.2 Suggestions for one-week projects