Math 6610 • Analysis of Numerical Methods I

{Convergence} Consider the one-dimensional heat equation initial
boundary value problem
\[
  u_t=u_{xx},\quad x\in[0,1],\quad t\geq0,\quad
  u(x,0)=f(x),\quad u(0,t)=u(1,t)=0 \tag{7}
\]
and the discretization
\[
  x_m=mh,\qquad t_n=nk,\qquad \text{and}\qquad U_m^n\approx u(x_m,t_n)
  \tag{8}
\]
where
\[
  U_m^{n+1}=U_m^n+r\left(U_{m-1}^n-2U_m^n+U_{m+1}^n\right) \tag{9}
\]
and \(r=k/h^2\) is the grid constant. Define what it means that this method is
convergent, say for what values of \(r\) it is convergent, and show that your
statement is correct.