Math 6610 • Analysis of Numerical Methods I

{Higher Order Methods}

Consider the following problem for \(v(x)\) on \([0,1]\):
\[
  v''(x)=f(x), \tag{2}
\]
with \(v(0)=v(1)=0\). Let \(N\cdot h=1\) and define \(x_j=j\cdot h\) for
\(j=0,1,\ldots,N\). The finite-difference scheme
\[
  \Delta_h U_j^h\equiv
  \frac{1}{h^2}\left(U_{j-1}^h-2U_j^h+U_{j+1}^h\right)
  =
  f_j^h,
\]
for \(j=1,\ldots,N-1\) with \(U_0^h=U_N^h=0\) gives values \(U_j^h\) that approximate
\(v(x_j)\) with an error of \(O(h^2)\). Here the superscript \(h\) is used to indicate
the grid size for the solution. Show how to use this method to find a numerical
solution \(W_j\) whose values approximate \(v(x_j)\) with an error of \(O(h^4)\) and a
numerical solution \(Y_j\) whose values approximate \(v(x_j)\) with an error of
\(O(h^6)\).