I am attaching a solved example on initial boundary value problem. I have not understood the step underlined in red. I know it is given that
$\frac{\partial u}{\partial t}(x,0)=0$
But I have not understood how the underlined step follows from this. $u^{-1}_{i}$ would be referring to one level lower than the lowest level indicated in the figure. Why should values at these nodes be equal to those corresponding at one level above the lowest indicated in the figure by $\frac{\partial u}{\partial t}(x,0)=0$ ? Is there a physical interpretation to the fact that if partial derivative at those nodes is zero, then the values at next+1 level are same? Request help clarify.
The idea is to use the centered difference formula for the first derivative, rather than the forward difference formula. That is, you approximate $u_t(x,0)$ by $\frac{u(x,\Delta t)-u(x,-\Delta t)}{2\Delta t}$. To make sense of this, you have to go back in time a little bit, which is where $u^{-1}$ comes in. The reason to do this is to keep the second order convergence rate of the temporal discretization scheme. With the forward difference, the dominant error comes from the less accurate discretization of the boundary condition, and so the convergence rate is spoiled.
You can also use $u^{-1}$ to evaluate $u_{tt}(x,0)$ using the three point centered difference formula, which enters into the discretization of the PDE rather than the discretization of the boundary condition.