PDE Tutorial#
Welcome to the PDE tutorial.
Homesite: https://jonshiach.github.io/files/notes/finite_difference_methods_notes.pdf
Course: Numerical Methods for Partial Differential Equations II
Recordings:
Contents#
The Finite Difference Method
- 1. Partial Differential Equations
- 2. Finite-Difference Approximations
- 3. Elliptic Partial Differential Equations
- 4. octave
- 5. pystencils
- 6. Introduction to JAX
- 7. finitediffx
- 8. kernex
- 9.
py-pdetutorial - 10.
findifftutorial - 11. Symbolic
findiff - 12. Solve with
findiff - 13.
AdvectionPer.m - 14.
DiffusionCrankNic.m - 15. Advection pystencils example
- 16. 1D heat diffusion
The Finite Volume Method
The Mimetic Finite Difference Method
Simulating Fluid Flows Using Python
- 17. Introduction to Computational Fluid Dynamics
- 18. Finite Differences - First Derivative
- 19. Polynomial derivative estimation
- 20. Finite differences in Python
- 21. Second derivative - 1D heat conduction
- 22. Introduction to Finite Volume Method
- 23. CFD Relevant Python Statements | While | For
- 24. Conduction Visualization Python Script
- 25. Generalized FVM Formulation
- 26. 0
Computers, Waves, Simulations
- 27. First Derivative
- 28. Second Derivative
- 29. High-Order Taylor Operators
- 30. Acoustic Waves 1D I
- 31. Acoustic Waves 1D II
- 32. Acoustic Waves 1D - Optimal Operators
- 33. Acoustic Waves 2D - Heterogeneous case
- 34. Acoustic Waves 2D - Homogeneous case
- 35. Advection equation 1D
- 36. Grid-Staggering Elastic 1D
- 37. The Pseudo-Spectral Method - Numerical Derivatives based on a Derivative Matrix
- 38. The Pseudo-Spectral Method - Elastic Wave Equation 1D
- 39. The Pseudo-Spectral Method - Acoustic Waves in 1D
- 40. The Pseudo-Spectral Method - Acoustic Waves in 2D
- 41. The Finite Element Method - Static Elasticity
- 42. The Finite Element Method - Elastic Wave Equation 1D
- 43. The Spectral Element Method - Interpolation with Lagrange Polynomials
- 44. The Spectral Element Method - Numerical Integration - The Gauss-Lobatto-Legendre approach
- 45. Spectral Element Method - Elastic Wave Equation 1D, Heterogeneous case
- 46. Spectral Element Method - Elastic Wave Equation 1D - Homogeneous Case
Differential equations and Fourier analysis
- 47. Fourier series: supplemental material
- 48. Polynomial interpolation: Lagrange interpolation
- 49. Introduction
- 50. Methods
- 51. Numerical solution of ordinary differential equations: High order Runge-Kutta methods
- 52. Runge-Kutta Methods
- 53. Finite difference methods for two-point value problems
- 54. \begin{align} -\partial^+\partial^- U_i
- 55. \begin{align} \dfrac{1}{h^2} \underbrace{ \begin{bmatrix} -1 & 2 & -1 & & & \ & -1 & 2 & -1 & & \ & & -1 & 2 & -1 & & \ & & & \ddots &\ddots & \ddots & \ & & & & -1 & 2 & -1 \end{bmatrix} }{\widetilde{A}} \underbrace{ \begin{bmatrix} U_0 \ U_1 \ U_2 \ U_3 \ \vdots \ U{N} \end{bmatrix} }_{\widetilde{U}}
- 56. \begin{align} \dfrac{1}{h^2} \underbrace{ \begin{bmatrix} h^2 \ -1 & 2 & -1 & & & \ & -1 & 2 & -1 & & \ & & -1 & 2 & -1 & & \ & & & \ddots &\ddots & \ddots & \ & & & & -1 & 2 & -1 \ & & & & & & h^2 \end{bmatrix} } \underbrace{ \begin{bmatrix} U_0 \ U_1 \ U_2 \ U_3 \ \vdots \ U{N-1} \ U_{N} \end{bmatrix} }_
- 57. Numerical integration: Part II
- 58. General construction of quadrature rules
- 59. Constructing quadrature rules on a single interval
- 60. Composite quadrature rules
- 61. Numerical integration: Part IV
- 62. Newton-Cotes formulas
- 63. Gauß quadrature
- 64. Introduction to TMA4125
- 65. Taylor-expansions
- 66. Some other useful results
- 67. A simple epidemic model for Covid 19 and its numerical solution
- 68. Numerical solution of ordinary differential equations: One step methods
- 69. One Step Methods
- 70. Introduction to TMA4125
- 71. Taylor-expansions
- 72. Some other useful results
- 73. Polynomial interpolation: Newton interpolation
- 74. Numerical integration
- 75. Introduction
- 76. Quadrature based on polynomial interpolation.
- 77. Degree of exactness and an estimate of the quadrature error
- 78. Polynomial interpolation: Error theory
- 79. Theory
- 80. Numerical solution of ordinary differential equations: Euler’s and Heun’s method
- 81. Introduction: Whetting your appetite
- 82. Euler’s method
- 83. Heun’s method
- 84. Applying Heun’s and Euler’s method
- 85. Bibliography
- 86. Numerical methods for the heat equation
Finite Difference Computing with PDEs
End Matter