The one$-$dimensional linear wave equation can be written as

$$u$/$t$=$c $(^2$ u$)/($x$^2)$

where u $=$ u$($x$,$t$)$ is a second$-$order in time and second$-$order in space finite difference scheme and u is given by

where $\backslash$lambda $=($c $$t$)/$x

The stability condition for this scheme is the Courant Friedrichs Lewy $($CFL$)$ condition, hence $\backslash$lambda $<=1,$

where c $=$x$/$t is the Courant number. In order to reduce the computational time, the $\backslash$Delta t is usually kept as large as possible, hence reducing the number of time steps taken.

The problem is furthermore normalized such that an approximation error of e$1$ for a discretization size $\backslash$Delta x$=1$ is obtained. You may assume you are using a computer with a $64-$bit processor, for which you can expect a floating point accuracy $1016.$

If we choose $\backslash$lambda $=0\mathrm{.}01.$ What is the expected error when computing $($d$^2$ u$)/($x$^2)$ for the above discretization? $[$ans$1]$

Subsequently, you perform a mesh convergence study, and hence you refine the grid resolution. What is the smallest value of $\backslash$Delta x you can afford before you encounter errors when computing $($d$^2$ u$)/($x$^2)$ due to the machine precision? $[$ans$2]$

Having performed the tests for the one$-$dimensional problem, you decide to perform the simulations on a three$-$dimensional problem with $\backslash$lambda $=0\mathrm{.}01$ for a domain of unit size in each dimension and periodic boundary conditions, and $5$ seconds of physical simulation time are needed for the problem. Considering an $800$ MFLOPS $($mega floating point operations per second$)$ computational power and that each discretization point requires $5$ floating$-$point operations per time step, what is the total run time for the simulation, to the nearest hour? $[$ans$3]$