Finite Difference Computing with Exponential Decay Models

Contents:

• Finite Difference Computing with Exponential Decay Models
• Preface
• Algorithms and implementations
  • 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
    • Mathematical derivation of finite difference formulas
    • Compact operator notation for finite differences
• Implementation
  • Computer language: Python
  • Making a solver function
  • Integer division
  • Doc strings
  • Formatting numbers
  • Running the program
  • Plotting the solution
  • Verifying the implementation
    • Running a few algorithmic steps by hand
  • Computing the numerical error as a mesh function
  • Computing the norm of the error mesh function
    • Scalar computing
  • Experiments with computing and plotting
    • Combining plot files
    • Plotting with SciTools
  • Memory-saving implementation
• Exercises
  • Exercise 1.1: Define a mesh function and visualize it
    • Remarks
  • Problem 1.2: Differentiate a function
  • Problem 1.3: Experiment with divisions
  • Problem 1.4: Experiment with wrong computations
  • Problem 1.5: Plot the error function
  • Problem 1.6: Change formatting of numbers and debug
• Analysis
  • Experimental investigations
    • Discouraging numerical solutions
    • Detailed experiments
  • Stability
    • Exact numerical solution
    • Stability properties derived from the amplification factor
  • Accuracy
    • Visual comparison of amplification factors
    • Series expansion of amplification factors
    • The ratio of numerical and exact amplification factors
    • The global error at a point
    • Integrated error
    • Truncation error
    • Consistency, stability, and convergence
  • Various types of errors in a differential equation model
    • Model errors
    • Data errors
    • Discretization errors
    • Rounding errors
    • Discussion of the size of various errors
  • Exercises
    • Problem 2.1: Visualize the accuracy of finite differences
    • Problem 2.2: Explore the \(\theta\)-rule for exponential growth
    • Problem 2.3: Explore rounding errors in numerical calculus
• Generalizations
  • 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
    • Computing convergence rates
    • Extension to systems of ODEs
  • General first-order ODEs
    • Generic form of first-order ODEs
    • The \(\theta\)-rule
    • An implicit 2-step backward scheme
    • Leapfrog schemes
    • The 2nd-order Runge-Kutta method
    • A 2nd-order Taylor-series method
    • The 2nd- and 3rd-order Adams-Bashforth schemes
    • The 4th-order Runge-Kutta method
    • The Odespy software
    • Example: Runge-Kutta methods
    • Example: Adaptive Runge-Kutta methods
  • Exercises
    • Exercise 3.1: Experiment with precision in tests and the size of \(u\)
    • Exercise 3.2: Implement the 2-step backward scheme
    • Exercise 3.3: Implement the 2nd-order Adams-Bashforth scheme
    • Exercise 3.4: Implement the 3rd-order Adams-Bashforth scheme
    • Exercise 3.5: Analyze explicit 2nd-order methods
    • Project 3.6: Implement and investigate the Leapfrog scheme
    • Problem 3.7: Make a unified implementation of many schemes
• Models
  • Scaling
    • Dimensionless variables
    • Dimensionless numbers
    • A scaling for vanishing initial condition
  • Evolution of a population
    • Exponential growth
    • Logistic growth
  • Compound interest and inflation
  • Newton’s law of cooling
  • Radioactive decay
    • Deterministic model
    • Stochastic model
    • Relation between stochastic and deterministic models
    • Generalization of the radioactive decay modeling
  • Chemical kinetics
    • Irreversible reaction of two substances
    • Reversible reaction of two substances
    • Irreversible reaction of two substances into a third
    • A biochemical reaction
  • Spreading of diseases
  • Predator-prey models in ecology
  • Decay of atmospheric pressure with altitude
    • The general model
    • Multiple atmospheric layers
    • Simplifications
  • Compaction of sediments
  • Vertical motion of a body in a viscous fluid
    • Overview of forces
    • Equation of motion
    • Terminal velocity
    • A Crank-Nicolson scheme
    • Physical data
    • Verification
    • Scaling
  • Viscoelastic materials
  • Decay ODEs from solving a PDE by Fourier expansions
  • Exercises
    • Problem 4.1: Radioactive decay of Carbon-14
    • Exercise 4.2: Derive schemes for Newton’s law of cooling
    • Exercise 4.3: Implement schemes for Newton’s law of cooling
    • Exercise 4.4: Find time of murder from body temperature
    • Exercise 4.5: Simulate an oscillating cooling process
    • Exercise 4.6: Simulate stochastic radioactive decay
    • Problem 4.7: Radioactive decay of two substances
    • Exercise 4.8: Simulate a simple chemical reaction
    • Exercise 4.9: Simulate an \(n\)-th order chemical reaction
    • Exercise 4.10: Simulate a biochemical process
    • Exercise 4.11: Simulate spreading of a disease
    • Exercise 4.12: Simulate predator-prey interaction
    • Exercise 4.13: Simulate the pressure drop in the atmosphere
    • Exercise 4.14: Make a program for vertical motion in a fluid
    • Project 4.15: Simulate parachuting
    • Exercise 4.16: Formulate vertical motion in the atmosphere
    • Exercise 4.17: Simulate vertical motion in the atmosphere
    • Problem 4.18: Compute \(y=|x|\) by solving an ODE
    • Problem 4.19: Simulate fortune growth with random interest rate
    • Exercise 4.20: Simulate a population in a changing environment
    • Exercise 4.21: Simulate logistic growth
    • Exercise 4.22: Rederive the equation for continuous compound interest
    • Exercise 4.23: Simulate the deformation of a viscoelastic material
• Scientific software engineering
  • Implementations with functions and modules
    • Mathematical problem and solution technique
    • A first, quick implementation
    • A more decent program
    • Prefixing imported functions by the module name
    • Implementing the numerical algorithm in a function
    • Do not have several versions of a code
    • Making a module
    • Example on extending the module code
    • Documenting functions and modules
    • Logging intermediate results
  • User interfaces
    • Command-line arguments
    • Positional command-line arguments
    • Option-value pairs on the command line
    • Creating a graphical web user interface
  • Tests for verifying implementations
    • Doctests
    • Unit tests and test functions
    • Test function for the solver
    • Test function for reading positional command-line arguments
    • Test function for reading option-value pairs
    • Classical class-based unit testing
  • Sharing the software with other users
    • Organizing the software directory tree
    • Publishing the software at GitHub
    • Downloading and installing the software
  • Classes for problem and solution method
    • The problem class
    • The solver class
    • Improving the problem and solver classes
  • Automating scientific experiments
    • Available software
    • The results we want to produce
    • Combining plot files
    • Running a program from Python
    • The automating script
    • Making a report
    • Publishing a complete project
  • Exercises
    • Problem 5.1: Make a tool for differentiating curves
    • Problem 5.2: Make solid software for the Trapezoidal rule
    • Problem 5.3: Implement classes for the Trapezoidal rule
    • Problem 5.4: Write a doctest and a test function
    • Problem 5.5: Investigate the size of tolerances in comparisons
    • Exercise 5.6: Make use of a class implementation
    • Problem 5.7: Make solid software for a difference equation
• References

Index

• Index