Wave Equation in two Dimensions

If we assume that all values at times $t=j$ and $t=j-1$ are known, the only unknown variable is $u_{i,j+1}$ and the last equation yields thus an explicit scheme for updating this quantity. We have thus an explicit finite difference scheme for computing the wave function $u$. The only additional complication in our case is the initial condition given by the first derivative in time, namely $\partial u/\partial t|_{t=0}=0$. The discretized version of this first derivative is given by $$\begin{equation*} u_t\approx \frac{u(x_i,t_j+\Delta t)-u(x_i,t_j-\Delta t)}{2\Delta t}, \end{equation*}$$ and at $t=0$ it reduces to $$\begin{equation*} u_t\approx \frac{u_{i,+1}-u_{i,-1}}{2\Delta t}=0, \end{equation*}$$ implying that $u_{i,+1}=u_{i,-1}$.

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