Finite difference methods for wave motion¶
Contents:
- Finite difference methods for wave motion
- Simulation of waves on a string
- Verification
- Implementation
- Vectorization
- Exercises
- Generalization: reflecting boundaries
- 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
- 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: Explain why numerical noise occurs
- Exercise 10: Investigate harmonic averaging in a 1D model
- Problem 11: Implement open boundary conditions
- Exercise 12: Implement periodic boundary conditions
- Exercise 13: Compare discretizations of a Neumann condition
- Exercise 14: Verification by a cubic polynomial in space
- Analysis of the difference equations
- Finite difference methods for 2D and 3D wave equations
- Implementation
- Using classes to implement a simulator
- Exercises
- Applications of wave equations
- Spherical waves
- The linear shallow water equations
- Waves in blood vessels
- Electromagnetic waves
- Exercises
- Exercise 19: Simulate waves on a non-homogeneous string
- Exercise 20: Simulate damped waves on a string
- Exercise 21: Simulate elastic waves in a rod
- Exercise 22: Simulate spherical waves
- 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
- References