Implementing Numerical Integration

[Tangent87]

Say we have some partial differential equation for u(x,t) and we devise a numerical method to solve it. For example if we say uⁿ_m is the solution at x=hm and t=kn with m=0,...,M and n=0,...,N where h and k are the step-lengths, and then our method is something like:

$$u_m^{n+1}=u_m^{n-1}+2u_{m+1}^n-u_m^n-3u_{m-1}^n+k(u_{m+2}^n-u_{m-2}^n)$$

and we're given u(x,0) and u(x,k) so we can work out u⁰_m and u¹_m for all m. We then stick n=1 into the recurrence in a bid to work out u²_m for all m. But what happens if we stick m=0 or 1 now into the recurrence do we just let u¹_m-2 and u¹_m+2 be equal to 0 or is the recurrence just not defined for m=0 or 1? In which case how do we find u²₀ say?

 

[Tangent87]

Ok I've figured out that if we use boundary conditions such as u(0,t)=u(1,t) for all t then *I think* the recurrence relation will set us up with a set of simultaneous equations, however I have no idea how we can program to solve this?