Introduction to computing with finite difference methods

Contents:

• Introduction to computing with finite difference methods
• 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 as a mesh function
  • Computing the norm of the numerical error
  • Plotting solutions
  • Creating command-line interfaces
  • Creating a graphical web user interface
  • Computing convergence rates
  • Memory-saving implementation
• Software engineering
  • Making a module
  • Prefixing imported functions by the module name
  • Doctests
  • Unit testing with nose
  • Classical class-based unit testing
  • Implementing simple problem and solver classes
  • Improving the problem and solver classes
• Performing scientific experiments
  • Software
  • Combining plot files (2)
  • Interpreting output from other programs
  • Making a report
  • Publishing a complete project
• Exercises and Problems
  • Exercise 1: Derive schemes for Newton’s law of cooling
  • Exercise 2: Implement schemes for Newton’s law of cooling
  • Exercise 3: Find time of murder from body temperature
  • Exercise 4: Experiment with integer division
  • Exercise 5: Experiment with wrong computations
  • Exercise 6: Plot the error function
  • Exercise 7: Compare methods for a given time mesh
  • Exercise 8: Change formatting of numbers and debug
  • Problem 9: Write a doctest
  • Problem 10: Write a nose test
  • Problem 11: Make a module
  • Exercise 12: Make use of a class implementation
  • Exercise 13: Generalize a class implementation
  • Exercise 14: Generalize an advanced class implementation
• Analysis of finite difference equations
  • Experimental investigation of oscillatory solutions
  • Exact numerical solution
  • Stability
  • Comparing amplification factors
  • Series expansion of amplification factors
  • The fraction of numerical and exact amplification factors
  • The global error at a point
  • Integrated errors
  • Truncation error
  • Consistency, stability, and convergence
• Exercises (1)
  • Exercise 15: Visualize the accuracy of finite differences \(u=e^{-at}\)
  • Exercise 16: Explore the \(\theta\)-rule for exponential growth
• Model extensions
  • Generalization: including a variable coefficient
  • Generalization: including a source term
  • Implementation of the generalized model problem
  • Verifying a constant solution
  • Verification via manufactured solutions
  • Extension to systems of ODEs
• General first-order ODEs
  • Generic form
  • The \(\theta\)-rule
  • An implicit 2-step backward scheme
  • Leapfrog schemes
  • The 2nd-order Runge-Kutta scheme
  • A 2nd-order Taylor-series method
  • The 2nd- and 3rd-order Adams-Bashforth schemes
  • 4th-order Runge-Kutta scheme
  • The Odespy software
  • Example: Runge-Kutta methods
  • Example: Adaptive Runge-Kutta methods
• Exercises (2)
  • Exercise 17: Experiment with precision in tests and the size of \(u\)
  • Exercise 18: Implement the 2-step backward scheme
  • Exercise 19: Implement the 2nd-order Adams-Bashforth scheme
  • Exercise 20: Implement the 3rd-order Adams-Bashforth scheme
  • Exercise 21: Analyze explicit 2nd-order methods
  • Problem 22: Implement and investigate the Leapfrog scheme
  • Problem 23: Make a unified implementation of many schemes
• Applications of exponential decay models
  • Scaling (1)
  • Evolution of a population
  • Compound interest and inflation
  • Radioactive Decay
  • Newton’s law of cooling
  • Decay of atmospheric pressure with altitude
  • Compaction of sediments
  • Vertical motion of a body in a viscous fluid
  • Decay ODEs from solving a PDE by Fourier expansions
• Exercises and Projects
  • Exercise 24: Simulate an oscillating cooling process
  • Exercise 25: Radioactive decay of Carbon-14
  • Exercise 26: Simulate stochastic radioactive decay
  • Exercise 27: Radioactive decay of two substances
  • Exercise 28: Simulate the pressure drop in the atmosphere
  • Exercise 29: Make a program for vertical motion in a fluid
  • Project 30: Simulate parachuting
  • Exercise 31: Formulate vertical motion in the atmosphere
  • Exercise 32: Simulate vertical motion in the atmosphere
  • Exercise 33: Compute \(y=|x|\) by solving an ODE
  • Exercise 34: Simulate growth of a fortune with random interest rate
  • Exercise 35: Simulate a population in a changing environment
  • Exercise 36: Simulate logistic growth
  • Exercise 37: Rederive the equation for continuous compound interest
• Bibliography

Indices and tables

• Index
• Module Index
• Search Page