Problem $2.$ Runge$-$Kutta Methods Applied to the Free Fall Problem

Write a Matlab script based on three functions to solve the free fall problem. Taking air resistance into

account, the velocity $v(t)($depending on time $t)$ of a ball with mass $m=0\mathrm{.}6lbm$ and cross$-$sectional area

$A=0\mathrm{.}503f{t}^2$ in free fall can be described by the initial value problem $($IVP$)$

$v^{}(t)=$ubrace$(g-\frac{C_{D}A}{2m}v(t{)}^2$ubrace${)}_{f(v)}in(0,6]$

$v(0)=0.$

Here, $v^{}(t)=d\frac{v}{d}t,g=32\mathrm{.}174f\frac{t}{s^2}$ is the gravity of Earth, $=0\mathrm{.}0765lb\frac{m}{f}{t}^3$ is the air density, and

$C_D=0\mathrm{.}47$ is the drag coefficient.

The exact solution to this IVP is given by

$v(t)=v_{}tanh(\frac{gt}{v_{}})$

where $v_{}=\sqrt[\phantom{2}]{2m\frac{g}{C_{D}A}}.$ Use this exact solution to determine the orders of approximation of the

Runge$-$Kutta methods RK$1-4$ mentioned below through numerical means.

$[RK1,c_1,=tf(v_k)],[v_{k+1},=v_k+c_1],[RK2,c_1,=tf(v_k)],[c_2,=tf(v_k+\frac{1}{2}{c}_1)],[v_{k+1},=v_k+c_2]$Compare the first$-$order $($RK$1,$ aka forward or explicit Euler method$),$ second$-$order $($RK$2),$ third$-$order

$($RK$3),$ and fourth$-$order $($RK$4)($explicit$)$ Runge$-$Kutta methods, which generally rely on breaking a single

time step down into multiple, incremental calculations to reduce overall error. To this end, do the following:

Write the following function that contains all four methods to advance the approximate solution by

one time step.

Here, method is either $1,2,3,$ or $4$ to choose whether to calculate the value at $v_{k+1}$ using the first$-,$

second$-,$ third$-,$ or fourth$-$order Runge$-$Kutta approximation, respectively, dt is the time step size, and

$m,g,$ rho, C$\_$D$,$ and A are the free fall parameters mentioned above. The current velocity vk is used to

calculate the value $vkp1=vk+$dots at the next time step, and the updated value is returned as the

single output. Note that although all four methods are included in a single function, only the result

generated by the method option will be calculated with a single function call.

To make this function more versatile, write the following nested function that evaluates the given

right$-$hand side function $f(v)$ of the ordinary differential equation.

function $[$out$]=f(vk)$

$\%$dotsStart your calculations with the time step size $t=1s,$ and halve $t$ successively nine times to gen$-$

erate ten calculations for each Runge$-$Kutta method. For each time step $($in each calculation associated

with one $($constant$)$ time step size $t),$ call the computeRungeKuttaSolution function repeatedly

to solve the given IVP up to $t_{\text{f}\text{i}\text{n}\text{a}\text{l}}=6s$ using each of the four numerical methods. Generate three plots

associated with the time step sizes $tin\{1,0\mathrm{.}5,0\mathrm{.}25\}.$ Each plot should consist of the exact solution

and piecewise linear interpolants $($MATLAB default$)$ of the four Runge$-$Kutta approximations.

Determine the order of approximations of the various Runge$-$Kutta methods in terms of the ratios of

the $($absolute values of the$)$ exact errors, as obtained from each of the four Runge$-$Kutta approxima$-$

tions. More precisely, determine the ratios of the exact errors from the previous time step size to the

current time step size at the times tin$\{1,3,5\}.$ Use a $1-$D $($or $2-$D$)$ array to store the necessary exact

error values for each of the four Runge$-$Kutta approximations at the given times, and output the results

to the command window in the following format $($the values shown are for illustration purposes only$)$:The output shall be generated by the following output function.

function outputErrorResults$($method$,$dt$,$err,maxIt$)$

$\%...$

Here, err is the array with the error values mentioned above, and maxIt is the maximum number of

time step size iterations $($in this case $10).$ Note also that in this function, $dt$ is an array that includes all

time step sizes considered in this problem whereas it is a scalar value in the computeRungeKutta$-$

Solution function $($consequently$,$ the computeRungeKuttaSolution function needs to be called

with only one element of the dt array$).$

By examining the error ratios shown above, explain how you can verify through numerical means

that the orders of approximation of the four different Runge$-$Kutta methods are given by