2.43 Generate sparse matrix for the tridiagonal representation of second difference operator in 1D

The second derivative $\frac{d^{2}u}{d{x}^2}$ is approximated by $\frac{u_{i-1}-2u_i+u_{i+1}}{h^2}$ where $h$ is the grid spacing. Generate the $A$ matrix that represent this operator for $n=4$ where $n$ is the number of internal grid points on the line

Mathematica numberOfInternalGridPoints = 4; n = 3*internalGridPoints; sp = SparseArray[{Band[{1,1}]->-2, Band[{1,2}]->1, Band[{2,1}]->1}, {n,n}] Out[103]= SparseArray[<34>,{12,12}] MatrixForm[sp]

$$\left(\begin{array}{cccccccccccc}-2& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& -2& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& -2& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& -2& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& -2& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& -2& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& -2& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& -2& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& -2& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -2& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -2\end{array}\right)$$

 

Matlab internalGridPoints = 4; n = 3*internalGridPoints; e = ones(n,1); sp = spdiags([e -2*e e], -1:1, n,n); full(sp)

ans = -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 1 -2

 

Maple interface(rtablesize=16): LinearAlgebra:-BandMatrix([1,-2,1],1,12,12);

$$\left[\begin{array}{cccccccccccc}-2& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& -2& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& -2& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& -2& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 1& -2& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& -2& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& -2& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 1& -2& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 1& -2& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -2& 1\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -2\end{array}\right]$$