Practical Numerical Methods with Python

The course will consist of six learning modules that are somewhat self-contained. Each one will be motivated by a problem that can be modeled by a differential equation (or system of DEs) and will build new concepts in numerical computing, new coding skills and ideas about analysis of numerical solutions.

The learning modules are roughly as follows:

The phugoid model of glider flight.

Described by a set of two nonlinear ordinary differential equations, the phugoid model motivates numerical time integration methods, and we will build it starting from an even simpler model (e.g., simple harmonic motion), building up to the full nonlinear model in 4 or 5 lessons on initial-value problems. Roughly, this module would include: a) Forward/backward differencing and Euler's method for simple harmonic motion; b) extension to the phugoid model; c) the midpoint method, convergence testing, local vs. global error; d) Runge-Kutta methods.

Space and Time—Introduction to finite-difference solutions of PDEs

Starting with the simplest model represented by a partial differential equation (PDE)—the linear convection equation in one dimension—, this module builds the foundation of using finite differencing in PDEs. (The module is based on the “CFD Python” collection, steps 1 through 4.) It also motivates CFL condition, numerical diffusion, accuracy of finite-difference approximations via Taylor series, consistency and stability, and the physical idea of conservation laws.

Riding the wave: convection problems.

Starting with the inviscid Burgers’ equation in conservation form and a 1D shock wave, cover a sampling of finite-difference convection schemes of various types: upwind, Lax-Friedrichs, Lax-Wendroff, MacCormack, then MUSCL (discussing limiters). Traffic-flow equation with MUSCL (from HyperPython). Reinforce concepts of numerical diffusion and stability, in the context of solutions with shocks. It will motivate spectral analysis of schemes, dispersion errors, Gibbs phenomenon, conservative schemes.

Spreading out: diffusion problems

Start with heat equation in 2D (first introduction of two-dimensional FD discretization). Introduce implicit methods: backward Euler, trapezoidal rule (Crank-Nicolson), backward-differentiation formula (BDF). Pattern formation models (reaction-diffusion). Theory content: A-stability (unconditional stability), L-stability. Fourier spectral methods and splitting.

Relax and hold steady: elliptic problems.

Laplace and Poisson equations (steps 9 and 10 of “CFD Python”), explained as systems relaxing under the influence of the boundary conditions and the Laplace operator; introducing the idea of pseudo-time and iterative methods. Linear solvers for PDEs : Jacobi’s method, slow convergence of low-frequency modes (matrix analysis of Jacobi), Jacobi as a smoother, Multigrid.

Boundaries take over: the boundary element method (BEM)

Weak and boundary integral formulation of elliptic partial differential equations; the free space Green's function. Boundary discretization: basis functions; collocation and Galerkin systems. The BEM stiffness matrix: dense versus sparse; matrix conditioning. Solving the BEM system: singular and near-singular integrals; Gauss quadrature integration.