Solving $u'' - 5u = 6$ with finite difference methods.

Question

I have a task:

For an equation: $$u'' - 5u = 6, x \in (0, 1)$$ $$u(0) = 0, u'(1) - 3u(1) = 1$$ construct a recurrence relation("scheme" in the original) with second order approximation on a two-point template and suggest a numerically stable method for solving it.

Two questions arise:
$1)$How can I approximate a second order derivative at all, using only two points? It seems that there is not enough information to capture the change of the first derivative.
$2)$Ignoring second order derivative, what methods can I try? I already tried to use "Tridiagonal matrix algorithm" but resulting matrix is not diagonally dominant and I don't know any other less straightforward tests for numerical stability of this method.

P.S.
I rarely look up any information on numerical methods in english so I'm not sure that scheme and two-point template is a widely accepted terminology. Following this thought there may be another mistakes of similar nature.

Answer 1

The following approach will satisfy the stated requirements. Whether is compatible with the spirit of the original text remains to be seen. You can solve your two point boundary value problem using the shooting method. Specifically, you should solve the initial value problem \begin{alignat}{3} u' &= v &\quad u(0) &= 0 \\ v' &= 5u + 6 &\quad v(0) &= v_0\\ \end{alignat} using, say, the trapezoidal rule with stepsize $h$. Here $v_0$ is a variable. Let $t \rightarrow u_h(t; v_0)$ and $t \rightarrow v_h(t; v_0)$ denote the grid functions constructed in this manner. The trapezoidal rule is a one step implicit Runge Kutta method which is second order accurate in the time step. You can now solve the nonlinear equation \begin{equation} v_h(1; v_0) - 3 u_h(1; v_0) = 0 \end{equation} with respect to $v_0$ using say the bisection algorithm, the secant method or perhaps even a hybrid of the two. The stability of the overall scheme can now be investigated in terms of the good properties of the two algorithms involved. You will inherit the second order accuracy from the Runge-Kutta method.

Personally, I believe that it is a bit of a stretch to ask a student to reinvent the shooting method from scratch, but obviously, I do not know your textbook and the situation.

I hope this helps.

Answer 2

Perhaps the task is to be interpreted as using on a uniform grid with step size $h=1/n$ $$ u''(x_k)+O(h^2)=\frac{u_{k+1}-2u_k+u_{k-1}}{h^2}=5u_k+6 $$ as linear recurrence for the inner nodes $k=1,…,n-1$, and translate the boundary conditions as $u_0=0$ and $$ u'(1)+O(h^2) %= \frac{u_n-u_{n-1}}h-\frac{h}4u''(x_{n-\frac12})+O(h^2) =\frac{3u_n-2u_{n-1}-u_{n-2}}{4h} =1+3u_n $$ to get an essentially 3-banded linear system that can be then solved by the linear method of your preference.