Example 1.

Use the standart five-point explicit method to solve a two-dimansional wave equation

$$u_{t,t}=c^2(u_{xx}+u_{yy}),\quad u=u(x,y,t)$$

on the rectangular domain $[0, L]\times[0, L]$ with Dirichlet boundary conditions. Other parameters are:

Space interval	$L$ =1
Space discretization step	$\triangle x=\triangle y=0.01$
Time discretization step	$\triangle t=0.0025$
Amount of time steps	$T=2000$
Initial condition	$u(x,y,0)=4x^2y(1-x)(1-y)$

Solution for two different times can be seen on Fig. ().

Figure: Numerical solution of the two-dimensional wave equation, shown on two different times.