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 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$3]$