This is still a preliminary version.
Simulation of waves on a string
Discretizing the domain
Uniform meshes
The discrete solution
Fulfilling the equation at the mesh points
Replacing derivatives by finite differences
Algebraic version of the PDE
Algebraic version of the initial conditions
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
Making a solver function
Verification: exact quadratic solution
Visualization: animating the solution
Visualization via SciTools
Making movie files
Skipping frames for animation speed
Visualization via Matplotlib
Running a case
The benefits of scaling
Vectorization
Operations on slices of arrays
Finite difference schemes expressed as slices
Verification
Efficiency measurements
Remark on the updating of arrays
Exercises
Exercise 1: Simulate a standing wave
Remarks
Exercise 2: Add storage of solution in a user action function
Exercise 3: Use a class for the user action function
Exercise 4: Compare several Courant numbers in one movie
Project 5: Calculus with 1D mesh functions
Generalization: reflecting boundaries
Neumann boundary condition
Discretization of derivatives at the boundary
Implementation of Neumann conditions
Index set notation
Alternative implementation via ghost cells
Idea
Implementation
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 model PDE 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 6: Find the analytical solution to a damped wave equation
Problem 7: Explore symmetry boundary conditions
Exercise 8: Send pulse waves through a layered medium
Exercise 9: Compare discretizations of a Neumann condition
Exercise 10: 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
Preliminary results
Numerical wave propagation
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
Discretizing the PDEs
Handling boundary conditions where is \( u \) known
Discretizing the Neumann condition
Implementation
Scalar computations
Domain and mesh
Solution arrays
Index sets
Computing the solution
Vectorized computations
Verification
Testing a quadratic solution
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
Efficiency
Migrating loops to Fortran
The Fortran subroutine
Building the Fortran module with f2py
How to avoid array copying
Efficiency
Migrating loops to C via Cython
Translating index pairs to single indices
The complete C code
The Cython interface file
Building the extension module
Efficiency
Migrating loops to C via f2py
Migrating loops to C++ via f2py
Using classes to implement a simulator
Exercises
Exercise 11: Check that a solution fulfills the discrete model
Project 12: Calculus with 2D/3D mesh functions
Exercise 13: Implement Neumann conditions in 2D
Exercise 14: Test the efficiency of compiled loops in 3D
Applications of wave equations
Waves on a string
Damping
External forcing
Modeling the tension via springs
Waves on a membrane
Elastic waves in a rod
The acoustic model for seismic waves
Anisotropy
Sound waves in liquids and gases
Spherical waves
The linear shallow water equations
Wind drag on the surface
Bottom drag
Effect of the Earth's rotation
Waves in blood vessels
Electromagnetic waves
Exercises
Exercise 15: Simulate waves on a non-homogeneous string
Exercise 16: Simulate damped waves on a string
Exercise 17: Simulate elastic waves in a rod
Exercise 18: Simulate spherical waves
Exercise 19: Explain why numerical noise occurs
Exercise 20: Investigate harmonic averaging in a 1D model
Problem 21: Implement open boundary conditions
Remarks
Exercise 22: Implement periodic boundary conditions
Problem 23: Earthquake-generated tsunami over a subsea hill
Problem 24: Earthquake-generated tsunami over a 3D hill
Problem 25: Investigate Matplotlib for visualization
Problem 26: Investigate visualization packages
Problem 27: Implement loops in compiled languages
Exercise 28: Simulate seismic waves in 2D
Project 29: Model 3D acoustic waves in a room
Project 30: Solve a 1D transport equation
Problem 31: General analytical solution of a 1D damped wave equation
Problem 32: General analytical solution of a 2D damped wave equation