# Finite Difference Numerical Solution to NL coupled PDEs

 P: 98 Are those the Shallow Water equations? Using a finite differencing technique; there are several ways to solve the problem: If you use an explicit technique (meaning an iterative [n+2] relies on [n+1] and [n]), I could recommend something called the: FTCS ---> (Forward Time Centered Space) Forward time: $f_{t} = (f(t+h) - f(t))/h = (f^{n+1}_{i} - f^{n}_{i})/h$ Space Centered: $f_{x} = (f(x+h) - f(x-h))/(2h) = (f^{n}_{i+1} - f^{n}_{i-1})/(2h)$ The idea is this... t 4|x . . . . x 3|x . . . . x 2|x . . . . x 1|o o o o o o ----------------------> x _-1 2 3 4 5 6 "o" represents the Initial conditions. "x" represents the Boundary conditions. "." represents an unknown. Represent your Nonlinear system of PDES such that: (time+1) = (time) && (Space+1) && (Space-1) Think about how you'd use this scheme to solve or the first "." on the bottom left. If you use this scheme explicitly then it will be subject to the CFL condition... you should look it up, this is what will determine the stability of the explicit FTCS technique (whether or not the solution will converge, or blow up). Not to confuse the issue further, but you can effectively ignore the CFL condition if you write an implicit FTCS technique. This will become a system of matrices that you need to solve. It's more complicated to write something like this. So I wouldn't advise it unless you feel like challenging yourself. The explicit technique is easier to start with.