Neumann Boundary Conditions using FTCS on the Heat Equation

[tlonster]

I am really confused with the concept of Neumann Boundary conditions. For the simple PDE

u_t=u_xx for the domain from 0<=x<=1

I'm trying to use a ghost point (maintain a second order scheme) for the Neumann Boundary condition u_x(0,t) = 0.

I understand that I can setup a scheme to calculate u(0,t) by

u(0,n+1) = (1-2r)u(0,n) + 2ru(1,n)

What are the u(0,n) and u(1,n) representative of?

I'm given u(x,0) = sin((3*∏)/2)(x+(1/3))

Any help would be appreciated to help me understand what those inputs actually are.
Thanks

 

[gtoguy97]

u(0,n) and u(1,n) are the values of the solution u at the two grid points x=0 and x=1 respectively at time step n. In other words, they are the numerical approximations of the PDE solution at the boundary points. The initial condition u(x,0) = sin((3*π)/2)(x+(1/3)) gives the values of the solution at x=0 and x=1 at time t=0. Thus, u(0,0) = sin(3π/2) + (1/3) and u(1,0) = sin(5π/2) + (1/3).