Implicit Scheme

We could also have used a midpoint approximation for the first derivative, resulting in $$\begin{equation*} u_t\approx \frac{u(x_i,t_j+\Delta t)-u(x_i,t_j-\Delta t)}{2\Delta t}, \end{equation*}$$ with a truncation error $O(\Delta t^2)$. Here we will stick to the backward formula and come back to the latter below. For the second derivative we use however $$\begin{equation*} u_{xx}\approx \frac{u(x_i+\Delta x,t_j)-2u(x_i,t_j)+u(x_i-\Delta x,t_j)}{\Delta x^2}, \end{equation*}$$ and define again $\alpha = \Delta t/\Delta x^2$.

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