Finite difference methods for wave motion

Contents:

• Finite difference methods for wave motion
• 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 (1)
  • A slightly generalized model problem
  • Using an analytical solution of physical significance
  • Manufactured solution
  • Constructing an exact solution of the discrete equations
• Implementation (1)
  • Making a solver function
  • Verification: exact quadratic solution
  • Visualization: animating the solution
  • Running a case
  • The benefits of scaling
• Vectorization
  • Operations on slices of arrays
  • Finite difference schemes expressed as slices
  • Verification (2)
  • Efficiency measurements
• Exercises (1)
  • Exercise 1: Simulate a standing wave
  • 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
• 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 (2)
  • 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
• 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 (3)
  • Scalar computations
  • Vectorized computations
  • Verification (3)
• Migrating loops to Cython
  • Declaring variables and annotating the code
  • Visual inspection of the C translation
  • Building the extension module (1)
  • 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 (2)
• Migrating loops to C via f2py
  • Migrating loops to C++ via f2py
• Using classes to implement a simulator
• Exercises (3)
  • Exercise 10: Check that a solution fulfills the discrete model
  • Project 11: Calculus with 2D/3D mesh functions
  • Exercise 12: Implement Neumann conditions in 2D
  • Exercise 13: 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
  • Wind drag on the surface
  • Bottom drag
  • Effect of the Earth’s rotation
• Waves in blood vessels
• Electromagnetic waves
  • Exercises (4)
• Exercise 14: Simulate waves on a non-homogeneous string
• Exercise 15: Simulate damped waves on a string
• Exercise 16: Simulate elastic waves in a rod
• Exercise 17: Simulate spherical waves
• Exercise 18: Explain why numerical noise occurs
• Exercise 19: Investigate harmonic averaging in a 1D model
• Problem 20: Implement open boundary conditions
• Problem 21: Earthquake-generated tsunami over a subsea hill
• Problem 22: Earthquake-generated tsunami over a 3D hill
• Problem 23: Investigate Matplotlib for visualization
• Problem 24: Investigate visualization packages
• Problem 25: Implement loops in compiled languages
• Exercise 26: Simulate seismic waves in 2D
• Project 27: Model 3D acoustic waves in a room
• Project 28: Solve a 1D transport equation
• Problem 29: General analytical solution of a 1D damped wave equation
• Problem 30: General analytical solution of a 2D damped wave equation

Indices and tables

• Index
• Module Index
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