I am looking for some direction for some theorems which may help me. I'm currently developing a new type of solver (so far only for the basic heat equation) one issue i'm facing is that as you may expect the solution quality degrades as t increases.
What I am looking for is a theorem (if there is such a thing, which I so far haven't found) that relates the solution $u(x,t)$ to the initial boundary condition of $u(x,0)$. The idea is then I can use this as a measure of the accuracy of my approximation at $u(x,t)$.
Assume the problem in the domain $x = [0,1],\, t=[0,1]$ of the form
$u_t =u_{xx} ;$
$u(x,0) = f(x);$
$u(0,t) = 0;$
$u(1,t) = 0;$
Thanks p.s. I am primarily and computer scientist so sorry if this question seems stupid!