An easy (but not so short) introduction to applied numerical computing

This repository contains the materials used for the applied numerical computing series in TuxRiders. The materials are obtained from the following sources with very minor changes and adaptation:

1. Lectures on scientific computing with Python by Robert Johansson
2. Practical Numerical Methods with Python by Lorena A. Barba, Ian Hawke, and Bernard Knaepen
3. An Easy Introduction to Finite Element Method and Variational Formulation by Mojtaba Barzegari

The mentioned repositores are embedded (and then partially modified) as git subtrees in this repository.

Table of Content

0. Introduction to Jupyter notebooks

Scientific programming with Python

1. Introduction to scientific computing with Python
2. Introduction to Python programming
3. Numpy - multidimensional data arrays
4. SciPy - Library of scientific algorithms for Python
5. matplotlib - 2D and 3D plotting in Python
6. Sympy - Symbolic algebra in Python
   • Extra: SymPy example - mass transfer in a cellular construct
7. Tools for high-performance computing (HPC) applications
8. Revision control software

Finite difference method for (partial) differential equations

9. Initial-value problems: solving nonlinear ordinary differential equations
   • Introducing the problem: the phugoid model of glider flight
   • Solving a single-equation model of oscillation using Euler's method
   • Solving a full phugoid model using vectorized Euler's method
   • Higher-order methods: modified Euler and Runge-Kutta
10. General aspects of numerical solution of partial differential equations
    • Discretizing equations: 1D linear convection
    • Stability and the CFL condition
    • Discretizing 2nd-order derivatives: 1-D Diffusion
    • Combining non-linear convection and diffusion: Burger's equation
11. Riding the wave: convection problems
    • Conservation laws and convection: solving a traffic flow model
    • Numerical schemes for hyperbolic partial differential equations
    • A better flux equation for the traffic flow model
    • Introducing finite volume method
12. Spreading out: diffusion problems
    • Parabolic partial differential equations: heat conduction
    • Implicit schemes for the 1D heat equation
    • 2D Heat conduction and finite difference: explicit schemes
    • 2D Heat conduction: implicit solution
    • Crank-Nicolson scheme
13. Elliptic problems
    • Iterative solution: Jacobi method for Laplace equation
    • 2D Poisson equation
    • Gauss-Seidel and successive over-relaxation (SOR) schemes
    • The method of conjugate gradients (CG)

Finite element method for (partial) differential equations

14. Quick overview of the finite element method
15. Function approximation by global functions
16. Function approximation by finite elements
    FreeFEM code - function approximation
17. Variational formulations with global basis functions
18. Variational formulations with finite elements
    FreeFEM code - variational formulation and solution of Poisson equation
19. Time-dependent variational forms
    FreeFEM code - time dependent diffusion equation with variable diffusion coefficient
20. Variational forms for systems of partial differential equations
    FreeFEM code - Stokes equations for lid cavity problem
21. Dealing with nonlinearity
    FreeFEM code - Picard method for nonlinear Poisson equation
    FreeFEM code - Picard-relaxation for nonlinear Poisson equation
    FreeFEM code - Newton method for nonlinear Poisson equation
22. Iterative solvers and preconditioning
    FreeFEM code - high-performance solution of Poisson equation using PETSc