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

We wish to adopt a second$-$order in time and second$-$order in space finite difference scheme, and a candidate is given by

The stability condition for this scheme is the Courant Friedrichs Lewy $($CFL$)$ condition, hence $0<=\backslash$sigma $<=1,$ where $\backslash$sigma $=$c$\backslash$Delta t $/\backslash$Delta x 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 $=1$ for a discretisation 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 $($i$.$e$.$ machine precision$)$ of $1016.$

If we choose a $\backslash$Delta x$=0\mathrm{.}001.$ What is the expected error when computing for the above discretisation? $[$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 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 $\backslash$Delta x$=103$ 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 flops$)$ capability, and that each discretisation point requires $8$ floating point operations per time step, what is the total run time for the simulation, to the nearest hour. $[$ans$3]$