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Implicit Scheme

In deriving the equations for the explicit scheme we started with the so-called forward formula for the first derivative, i.e., we used the discrete approximation \begin{equation*} u_t\approx \frac{u(x_i,t_j+\Delta t)-u(x_i,t_j)}{\Delta t}. \end{equation*} However, there is nothing which hinders us from using the backward formula \begin{equation*} u_t\approx \frac{u(x_i,t_j)-u(x_i,t_j-\Delta t)}{\Delta t}, \end{equation*} still with a truncation error which goes like O(\Delta t) .