Basic-Transient-PDEs

A collection of some common schemes to the unsteady advection equation.

This is a repo I created to understand better the numerical solution to transient PDEs in 1-D and 2-D. An attempt is made to make the codes self-explanatory. Numerical schemes explored will be Upwind, FTCS, Lax-Friedrich's, Lax-Wendroff and Crank-Nicholson (for 1-D Advection). A few diffusion problems have also been implemented. The motivation behind this repo is simple: to understand unsteady problems better with the help of some of the most basic PDEs one can encounter. The CFL number or the Courant number comes into picture for hyperbolic and parabolic problems, which basically governs aspects of stability and tendencies of the solution to swing into wild oscillations or die down with time. It must be less than 1 for the solution to be able to converge, however, this is not a sufficient condition in itself, rather a necessary condition. Complete and stable convergence will require more analysis (a popular one is the Von Neumann analysis) and hence, will also be dictated by the discretization principle itself and the order of discretization used. For a more heuristic understanding, the CFL condition is really about setting the right time step to ensure the flow within each cell is still evolving relative to the wave speed. For all problems, the initial condition set is a hat-function/step function