Preface
Vibration ODEs
Finite difference discretization
A basic model for vibrations
A centered finite difference scheme
Implementation
Making a solver function
Verification
Scaled model
Long time simulations
Using a moving plot window
Making animations
Using Bokeh to compare graphs
Using a line-by-line ascii plotter
Empirical analysis of the solution
Analysis of the numerical scheme
Deriving a solution of the numerical scheme
Exact discrete solution
Convergence
The global error
Stability
About the accuracy at the stability limit
Alternative schemes based on 1st-order equations
The Forward Euler scheme
The Backward Euler scheme
The Crank-Nicolson scheme
Comparison of schemes
Runge-Kutta methods
Analysis of the Forward Euler scheme
Energy considerations
Derivation of the energy expression
An error measure based on energy
The Euler-Cromer method
Forward-backward discretization
Equivalence with the scheme for the second-order ODE
Implementation
The velocity Verlet algorithm
Generalization: damping, nonlinear spring, and external excitation
A centered scheme for linear damping
A centered scheme for quadratic damping
A forward-backward discretization of the quadratic damping term
Implementation
Verification
Visualization
User interface
The Euler-Cromer scheme for the generalized model
Exercises and Problems
Problem 1: Use linear/quadratic functions for verification
Exercise 2: Show linear growth of the phase with time
Exercise 3: Improve the accuracy by adjusting the frequency
Exercise 4: See if adaptive methods improve the phase error
Exercise 5: Use a Taylor polynomial to compute \( u^1 \)
Exercise 6: Find the minimal resolution of an oscillatory function
Exercise 7: Visualize the accuracy of finite differences for a cosine function
Exercise 8: Verify convergence rates of the error in energy
Exercise 9: Use linear/quadratic functions for verification
Exercise 10: Use an exact discrete solution for verification
Exercise 11: Use analytical solution for convergence rate tests
Exercise 12: Investigate the amplitude errors of many solvers
Exercise 13: Minimize memory usage of a vibration solver
Exercise 14: Implement the solver via classes
Exercise 15: Interpret \( [D_tD_t u]^n \) as a forward-backward difference
Exercise 16: Use a backward difference for the damping term
Exercise 17: Analysis of the Euler-Cromer scheme
Wave equations
Simulation of waves on a string
Discretizing the domain
The discrete solution
Fulfilling the equation at the mesh points
Replacing derivatives by finite differences
Formulating a recursive algorithm
Sketch of an implementation
Verification
A slightly generalized model problem
Using an analytical solution of physical significance
Manufactured solution
Constructing an exact solution of the discrete equations
Implementation
Callback function for user-specific actions
The solver function
Verification: exact quadratic solution
Visualization: animating the solution
Running a case
Working with a scaled PDE model
Vectorization
Operations on slices of arrays
Finite difference schemes expressed as slices
Verification
Efficiency measurements
Remark on the updating of arrays
Exercises
Exercise 18: Simulate a standing wave
Exercise 19: Add storage of solution in a user action function
Exercise 20: Use a class for the user action function
Exercise 21: Compare several Courant numbers in one movie
Project 22: Calculus with 1D mesh functions
Generalization: reflecting boundaries
Neumann boundary condition
Discretization of derivatives at the boundary
Implementation of Neumann conditions
Index set notation
Verifying the implementation of Neumann conditions
Alternative implementation via ghost cells
Generalization: variable wave velocity
The model PDE with a variable coefficient
Discretizing the variable coefficient
Computing the coefficient between mesh points
How a variable coefficient affects the stability
Neumann condition and a variable coefficient
Implementation of variable coefficients
A more general PDE model with variable coefficients
Generalization: damping
Building a general 1D wave equation solver
User action function as a class
Pulse propagation in two media
Exercises
Exercise 23: Find the analytical solution to a damped wave equation
Problem 24: Explore symmetry boundary conditions
Exercise 25: Send pulse waves through a layered medium
Exercise 26: Explain why numerical noise occurs
Exercise 27: Investigate harmonic averaging in a 1D model
Problem 28: Implement open boundary conditions
Exercise 29: Implement periodic boundary conditions
Exercise 30: Compare discretizations of a Neumann condition
Exercise 31: Verification by a cubic polynomial in space
Analysis of the difference equations
Properties of the solution of the wave equation
More precise definition of Fourier representations
Stability
Numerical dispersion relation
Extending the analysis to 2D and 3D
Finite difference methods for 2D and 3D wave equations
Multi-dimensional wave equations
Mesh
Discretization
Implementation
Scalar computations
Vectorized computations
Verification
Using classes to implement a simulator
Exercises
Exercise 32: Check that a solution fulfills the discrete model
Project 33: Calculus with 2D mesh functions
Exercise 34: Implement Neumann conditions in 2D
Exercise 35: Test the efficiency of compiled loops in 3D
Applications of wave equations
Waves on a string
Waves on a membrane
Elastic waves in a rod
The acoustic model for seismic waves
Sound waves in liquids and gases
Spherical waves
The linear shallow water equations
Waves in blood vessels
Electromagnetic waves
Exercises
Exercise 36: Simulate waves on a non-homogeneous string
Exercise 37: Simulate damped waves on a string
Exercise 38: Simulate elastic waves in a rod
Exercise 39: Simulate spherical waves
Problem 40: Earthquake-generated tsunami over a subsea hill
Problem 41: Earthquake-generated tsunami over a 3D hill
Problem 42: Investigate Matplotlib for visualization
Problem 43: Investigate visualization packages
Problem 44: Implement loops in compiled languages
Exercise 45: Simulate seismic waves in 2D
Project 46: Model 3D acoustic waves in a room
Project 47: Solve a 1D transport equation
Problem 48: General analytical solution of a 1D damped wave equation
Problem 49: General analytical solution of a 2D damped wave equation
Diffusion equations
The 1D diffusion equation
The initial-boundary value problem for 1D diffusion
Forward Euler scheme
Backward Euler scheme
Sparse matrix implementation
Crank-Nicolson scheme
The \( \theta \) rule
The Laplace and Poisson equation
Extensions
Analysis of schemes for the diffusion equation
Properties of the solution
Example: Diffusion of a discontinues profile
Analysis of discrete equations
Analysis of the finite difference schemes
Analysis of the Forward Euler scheme
Analysis of the Backward Euler scheme
Analysis of the Crank-Nicolson scheme
Summary of accuracy of amplification factors
Exercise 50: Explore symmetry in a 1D problem
Exercise 51: Investigate approximation errors from a \( u_x=0 \) boundary condition
Exercise 52: Experiment with open boundary conditions in 1D
Exercise 53: Simulate a diffused Gaussian peak in 2D/3D
Exercise 54: Examine stability of a diffusion model with a source term
Diffusion in heterogeneous media
Stationary solution
Piecewise constant medium
Implementation
Diffusion equation in axi-symmetric geometries
Diffusion equation in spherically-symmetric geometries
Exercises
Exercise 55: Stabilizing the Crank-Nicolson method by Rannacher time stepping
Project 56: Energy estimates for diffusion problems
Staggered mesh discretization
The Euler-Cromer scheme on a standard mesh
The Euler-Cromer scheme on a staggered mesh
Implementation of the scheme on a staggered mesh
A staggered Euler-Cromer scheme for a generalized model
Exercises
Exercise 57: Use the forward-backward scheme with quadratic damping
Appendix: Useful formulas
Finite difference operator notation
Truncation errors of finite difference approximations
Finite differences of exponential functions
Finite differences of \( t^n \)
Appendix: Truncation error analysis
Overview of truncation error analysis
Abstract problem setting
Error measures
Truncation errors in finite difference formulas
Example: The backward difference for \( u'(t) \)
Example: The forward difference for \( u'(t) \)
Example: The central difference for \( u'(t) \)
Overview of leading-order error terms in finite difference formulas
Software for computing truncation errors
Truncation errors in exponential decay ODE
Truncation error of the Forward Euler scheme
Truncation error of the Crank-Nicolson scheme
Truncation error of the \( \theta \)-rule
Using symbolic software
Empirical verification of the truncation error
Increasing the accuracy by adding correction terms
Extension to variable coefficients
Exact solutions of the finite difference equations
Computing truncation errors in nonlinear problems
Truncation errors in vibration ODEs
Linear model without damping
Model with damping and nonlinearity
Extension to quadratic damping
The general model formulated as first-order ODEs
Truncation errors in wave equations
Linear wave equation in 1D
Finding correction terms
Extension to variable coefficients
1D wave equation on a staggered mesh
Linear wave equation in 2D/3D
Truncation errors in diffusion equations
Linear diffusion equation in 1D
Linear diffusion equation in 2D/3D
A nonlinear diffusion equation in 2D
Exercises
Exercise 58: Truncation error of a weighted mean
Exercise 59: Simulate the error of a weighted mean
Exercise 60: Verify a truncation error formula
Exercise 61: Truncation error of the Backward Euler scheme
Exercise 62: Empirical estimation of truncation errors
Exercise 63: Correction term for a Backward Euler scheme
Exercise 64: Verify the effect of correction terms
Exercise 65: Truncation error of the Crank-Nicolson scheme
Exercise 66: Truncation error of \( u'=f(u,t) \)
Exercise 67: Truncation error of \( [D_t D_tu]^n \)
Exercise 68: Investigate the impact of approximating \( u'(0) \)
Exercise 69: Investigate the accuracy of a simplified scheme
Appendix: Software engineering; wave equation model
A 1D wave equation simulator
Mathematical model
Numerical discretization
A solver function
Saving large arrays in files
Using savez to store arrays in files
Using joblib to store arrays in files
Using a hash to create a file or directory name
Software for the 1D wave equation
Making hash strings from input data
Avoiding rerunning previously run cases
Verification
Programming the solver with classes
Class Problem
Class Mesh
Class Function
Class Solver
Migrating loops to Cython
Declaring variables and annotating the code
Visual inspection of the C translation
Building the extension module
Calling the Cython function from Python
Migrating loops to Fortran
The Fortran subroutine
Building the Fortran module with f2py
How to avoid array copying
Migrating loops to C via Cython
Translating index pairs to single indices
The complete C code
The Cython interface file
Building the extension module
Migrating loops to C via f2py
Migrating loops to C++ via f2py
Exercises
Exercise 70: Make an improved numpy.savez function
References