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
Making a solver function
Verifying the implementation
Computing the numerical error as a mesh function
Computing the norm of the error mesh function
Experiments with computing and plotting
Memory-saving implementation
Exercises
Exercise 1: Define a mesh function and visualize it
Exercise 2: Differentiate a function
Exercise 3: Experiment with integer division
Exercise 4: Experiment with wrong computations
Exercise 5: Plot the error function
Exercise 6: Change formatting of numbers and debug
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
Exercise 7: Visualize the accuracy of finite differences
Exercise 8: 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
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 9: Experiment with precision in tests and the size of \( u \)
Exercise 10: Implement the 2-step backward scheme
Exercise 11: Implement the 2nd-order Adams-Bashforth scheme
Exercise 12: Implement the 3rd-order Adams-Bashforth scheme
Exercise 13: Analyze explicit 2nd-order methods
Problem 14: Implement and investigate the Leapfrog scheme
Problem 15: Make a unified implementation of many schemes
Applications of exponential decay models
Scaling
Evolution of a population
Compound interest and inflation
Newton's law of cooling
Radioactive decay
Chemical kinetics
Spreading of diseases
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
Exercise 16: Radioactive decay of Carbon-14
Exercise 17: Derive schemes for Newton's law of cooling
Exercise 18: Implement schemes for Newton's law of cooling
Exercise 19: Find time of murder from body temperature
Exercise 20: Simulate an oscillating cooling process
Exercise 21: Simulate stochastic radioactive decay
Exercise 22: Radioactive decay of two substances
Exercise 23: Simulate a simple chemical reaction
Exercise 24: Simulate an \( n \)-th order chemical reaction
Exercise 25: Simulate spreading of a disease
Exercise 26: Simulate a biochemical process
Exercise 27: Simulate the pressure drop in the atmosphere
Exercise 28: Make a program for vertical motion in a fluid
Project 29: Simulate parachuting
Exercise 30: Formulate vertical motion in the atmosphere
Exercise 31: Simulate vertical motion in the atmosphere
Exercise 32: Compute \( y=|x| \) by solving an ODE
Exercise 33: Simulate growth of a fortune with random interest rate
Exercise 34: Simulate a population in a changing environment
Exercise 35: Simulate logistic growth
Exercise 36: Rederive the equation for continuous compound interest
Summarizing multiple-choice questions
Exercise 37: Characterize a finite difference
Exercise 38: Characterize a finite difference
Exercise 39: What is the problem with this program?
Exercise 40: Is the solution correct?
Exercise 41: Is this a proper test function?
Exercise 42: Rewrite an expression with array arithmetics
Exercise 43: What is the truncation error?
Exercise 44: Recognize a programming language
Exercise 45: Recognize a programming language
Exercise 46: Recognize a programming language
Exercise 47: Recognize a programming language
Exercise 48: What is SymPy?
Exercise 49: Testing of code
Exercise 50: What kind of scheme is this?
Exercise 51: What kind of scheme is this?
Exercise 52: What kind of scheme is this?
Bibliography
Finite difference methods for partial differential equations (PDEs) employ a range of concepts and tools that can be introduced and illustrated by way simple ordinary differential equation (ODE) examples. The aim of the present document is to lay a foundation for understanding numerical methods for PDEs by first meeting the fundamental ideas in a simpler ODE setting. With the ODEs, the mathematical problems are kept as simple as possible (but no simpler!), allowing full focus on the understanding of key concepts and tools. The choice of ODE topics to be covered here is thus solely determined by what carries over to the world of numerical solution methods for PDEs.
Theory and practice are primarily illustrated by solving the very simple ODE \( u'=-au \), \( u(0)=I \), where \( a>0 \) is a constant, but we also address the more general model problem \( u'=-a(t)u + b(t) \) and the completely general, nonlinear problem \( u'=f(u,t) \). The following list of topics will be elaborated on.
sympy software
for symbolic computations