Please provide step$-$by$-$step calculations with explanation.

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

$\frac{de{l}^{2}u}{del{t}^2}=c^2\frac{de{l}^{2}u}{del{x}^2}$

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

$\frac{u_i^{n+1}-2u_i^n+u_i^{n-1}}{t^2}-c^2\frac{u_{i+1}^n-2u_i^n+u_{i-1}^n}{x^2}=0$

The stability condition for this scheme is the Courant Friedrichs Lewy $($CFL$)$ condition, hence $01,$ where $=c\frac{t}{}x$ is the Courant number. In order to reduce the

computational time, the $t$ is usually kept as large as possible, hence reducing the number of time steps taken.

The problem is furthermore normalised such that an approximation error of $=1$ for a discretisation size $x=1$ 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 ${10}^{-16}.$

If we choose a $x=0\mathrm{.}001.$ What is the expected error when computing $\frac{de{l}^{2}u}{del{x}^2}$ 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 $x$ you can afford before you encounter errors

when computing $\frac{de{l}^{2}u}{del{x}^2}$ 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 $x={10}^{-3}$ 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$3]$