Finite difference methods
A basic model for exponential decay
The Forward Euler scheme
Step 1: Discretizing the domain
Step 2: Fulfilling the equation at discrete time points
Step 3: Replacing derivatives by finite differences
Step 4: Formulating a recursive algorithm
The Backward Euler scheme
The Crank-Nicolson scheme
The unifying \( \theta \)-rule
Constant time step
Compact operator notation for finite differences
Implementation
Making a solver function
Function for computing the numerical solution
Integer division
Doc strings
Formatting of 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 numerical error
Scalar computing
Plotting solutions
Experiments with computing and plotting
Combining plot files
Plotting with SciTools
Memory-saving 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
Exercise 1: Visualize the accuracy of finite differences
Exercise 2: Explore the \( \theta \)-rule for exponential growth
Model extensions
Generalization: including a variable coefficient
Generalization: including a source term
Schemes
Implementation of the generalized model problem
Deriving the \( \theta \)-rule formula
The Python code
Coding of variable coefficients
Verifying a constant solution
Verification via manufactured solutions
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 ordinary Leapfrog scheme
The filtered Leapfrog scheme
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
Remark about using the \( \theta \)-rule in Odespy
Example: Adaptive Runge-Kutta methods
Exercises
Exercise 3: Experiment with precision in tests and the size of \( u \)
Exercise 4: Implement the 2-step backward scheme
Exercise 5: Implement the 2nd-order Adams-Bashforth scheme
Exercise 6: Implement the 3rd-order Adams-Bashforth scheme
Exercise 7: Analyze explicit 2nd-order methods
Problem 8: Implement and investigate the Leapfrog scheme
Problem 9: Make a unified implementation of many schemes
Applications of exponential decay models
Scaling
Evolution of a population
Compound interest and inflation
Radioactive Decay
Deterministic model
Stochastic model
Relation between stochastic and deterministic models
Newton's law of cooling
Decay of atmospheric pressure with altitude
Multiple atmospheric layers
Simplification: \( L=0 \)
Simplification: one-layer model
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
Decay ODEs from solving a PDE by Fourier expansions
Exercises
Exercise 10: Derive schemes for Newton's law of cooling
Exercise 11: Implement schemes for Newton's law of cooling
Exercise 12: Find time of murder from body temperature
Exercise 13: Simulate an oscillating cooling process
Exercise 14: Radioactive decay of Carbon-14
Exercise 15: Simulate stochastic radioactive decay
Exercise 16: Radioactive decay of two substances
Exercise 17: Simulate the pressure drop in the atmosphere
Exercise 18: Make a program for vertical motion in a fluid
Project 19: Simulate parachuting
Exercise 20: Formulate vertical motion in the atmosphere
Exercise 21: Simulate vertical motion in the atmosphere
Exercise 22: Compute \( y=|x| \) by solving an ODE
Exercise 23: Simulate growth of a fortune with random interest rate
Exercise 24: Simulate a population in a changing environment
Exercise 25: Simulate logistic growth
Exercise 26: Rederive the equation for continuous compound interest
Bibliography
Finite difference methods for partial differential equations (PDEs) employ a range of concepts and tools that can be introduced and illustrated in the context of simple ordinary differential equation (ODE) examples. This is what we do in the present document. By first working with ODEs, we keep the mathematical problems to be solved as simple as possible (but no simpler), thereby allowing full focus on understanding the key concepts and tools. The choice of topics in the forthcoming treatment of ODEs is therefore solely dominated by what carries over to numerical 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 generalized problem \( u'=-a(t)u + b(t) \) and the nonlinear problem \( u'=f(u,t) \). The following topics are introduced:
sympy software
for symbolic computationEverything we cover is put into a practical, hands-on context. All mathematics is translated into working computing codes, and all the mathematical theory of finite difference methods presented here is motivated from a strong need to understand strange behavior of programs. Two fundamental questions saturate the text: