Finite difference methods for first-order ODEs

Contents:

• Finite difference methods for first-order ODEs
• Finite difference methods
  • A basic model for exponential decay
  • The Forward Euler scheme
  • The Backward Euler scheme
  • The Crank-Nicolson scheme
  • The unifying \(\theta\)-rule
  • Constant time step
  • Compact operator notation for finite differences
• Implementation (1)
  • Making a solver function
  • Verifying the implementation
  • Computing the numerical error
  • Plotting solutions
  • Plotting with SciTools
  • Creating user interfaces
  • Computing convergence rates
  • Memory-saving implementation
• Software engineering
  • Making a module
  • Prefixing imported functions by the module name
  • Doctests
  • Unit testing with nose
  • Classical unit testing with unittest
  • Implementing simple problem and solver classes
  • Implementing more advanced problem and solver classes
• Performing scientific experiments
  • Interpreting output from other programs
  • Making a report
  • Publishing a complete project
• Exercises (1)
  • Exercise 1: Experiment with integer division
  • Exercise 2: Experiment with wrong computations
  • Exercise 3: Implement specialized functions
  • Exercise 4: Plot the error function
  • Exercise 5: Compare methods for a give time mesh
  • Exercise 6: Change formatting of numbers and debug
  • Exercise 7: Write a doctest
  • Exercise 8: Write a nose test
  • Exercise 9: Make a module
  • Exercise 10: Make use of a class implementation
• Analysis of the \(\theta\)-rule for a decay ODE
  • Discouraging numerical solutions
  • Experimental investigation of oscillatory solutions
  • Exact numerical solution
  • Stability
  • Comparing Amplification Factors
  • Series Expansion of Amplification Factors
  • Local error
  • Analytical comparison of schemes
  • The real (global) error at a point
  • Integrated errors
• Exercises (2)
  • Exercise 11: Explore the \(\theta\)-rule for exponential growth
  • Exercise 12: Summarize investigations in a report
  • Exercise 13: Plot amplification factors for exponential growth
• Model extensions
  • Generalization: including a variable coefficient
  • Generalization: including a source term
  • Implementation of the generalized model problem
  • Verification via trivial solutions
  • Verification via manufactured solutions
  • Extension to systems of ODEs
• General first-order ODEs
  • Generic form
  • The Odespy software
  • Example: Runge-Kutta methods
  • Example: Adaptive Runge-Kutta methods
  • Other schemes
• Exercises (3)
  • Exercise 14: Implement the 2-step backward scheme
  • Exercise 15: Implement the Leapfrog scheme
  • Exercise 16: Experiment with the Leapfrog scheme
  • Exercise 17: Analyze the Leapfrog scheme
  • Exercise 18: Implement the 2nd-order Adams-Bashforth scheme
  • Exercise 19: Implement the 3rd-order Adams-Bashforth scheme
  • Exercise 20: Generalize a class implementation
  • Exercise 21: Generalize an advanced class implementation
  • Exercise 22: Make a unified implementation of many schemes
  • Exercise 23: Analyze explicit 2nd-order methods
• Applications of exponential decay models
  • Evolution of a population
  • Compound interest and inflation
  • Radioactive Decay
  • Newton’s law of cooling
  • Decay of atmospheric pressure with altitude
  • Vertical motion of a body in a viscous fluid
  • Decay ODEs from solving a PDE by Fourier expansions
• Exercises (4)
  • Exercise 24: Simulate the pressure drop in the atmosphere
  • Exercise 25: Make a program for vertical motion in a fluid
  • Exercise 26: Plot forces acting in vertical motion in a fluid
  • Exercise 27: Simulate a free fall of a parachute jumper
  • Exercise 28: Simulate a complete parachute jump
  • Exercise 29: Simulate a rising ball in water
  • Exercise 30: Radioactive decay of Carbon-14
  • Exercise 31: Radioactive decay of two substances
  • Exercise 32: Find time of murder from body temperature
  • Exercise 33: Simulate an oscillating cooling process
  • Exercise 34: Compute \(y=|x|\) by solving an ODE
  • Exercise 35: Simulate growth of a fortune with random interest rate
  • Exercise 36: Simulate sudden environmental changes for a population
  • Exercise 37: Simulate oscillating environment for a population
  • Exercise 38: Simulate logistic growth
  • Exercise 39: Rederive the equation for continuous compound interest

Indices and tables

• Index
• Module Index
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