P: 8 Hi, I want to numerically model the advection equation using the Crank-Nicolson scheme. Yes, I know that it is highly oscillatory but that is the point of the exercise as I want to highlight this. The problem I'm having is how do I apply the BC for grid point N in the scheme. The advection equation only needs one boundary condition at point 0 in the domain, but becuase of the centred space disretisation the scheme requires an artificial boundary condition at the other end. The basic equation is $$\-\frac{\sigma}{4}f_{i-1}^{n+1}+f_{i}^{n+1}+\frac{\sigma}{4}f_{i+1}^{n+1}=\frac{\sigma}{4}f_{i-1}^{n}+f_{i}^{n}-\frac{\sigma}{4}f_{i+1}^{n}$$ So say I want the value at N+1 to be the same as N, that requires $$\left.\frac{\partial f}{\partial x}\right|_{N}=\frac{f_{N+1}^{n}-f_{N-1}^{n}}{2\triangle x}=0$$ and hence $$f_{N+1}=f_{N-1}$$ So if we sub that into the main scheme we get $$f_{N}^{n+1}=f_{N}^{n}$$ So according to this the final grid point always remains at the initial condictions, which is clearly wrong. Does anyone know what is wrong with my assumptions? Thanks for any info.