Investigate various aspects of the $Ly=f\left(y\right)$ problem with $L$ a linear operator and $f$ nonlinear. Algorithms will be applied to the Van der Pol oscillator for $t\in [0,250]$ with $\mu =1$,

Plots of the solution are typically presented in phase space as (x(t),x'(t)).

1Track 1

1. Rewrite the second-order ODE as a system of first-order equations ${𝒚}^{\text{'}}=𝒇(t,𝒚)$.

2. Carry out a convergence study when applying the single-step Runge-Kutta method

   ${\displaystyle {𝒚}_{n+1}={𝒚}_n+\frac{1}{6}({𝒌}_1+2{𝒌}_2+2{𝒌}_3+{𝒌}_4),}$ ${\displaystyle \begin{array}{c}{𝒌}_1=h𝒇(t_n,{𝒚}_n)\\ {𝒌}_2=h𝒇(t_n+\frac{1}{2}h,{𝒚}_n+\frac{1}{2}{𝒌}_1)\\ {𝒌}_3=h𝒇(t_n+\frac{1}{2}h,{𝒚}_n+\frac{1}{2}{𝒌}_2)\\ {𝒌}_4=h𝒇(t_n+h,{𝒚}_n+{𝒌}_3)\end{array}}$

3. Carry out a convergence study when applying the Adams-Bashforth of order 4.

2Track 2

1. Rewrite the second-order ODE as a system of first-order equations ${𝒚}^{\text{'}}=𝒇(t,𝒚)$.

2. Use a symbolic package (e.g., Mathematica) to verify the fourth-order theoretical accuracy of the Runge-Kutta method from Track 1.

3. Carry out a convergence study using Adams-Moulton of fourth order. This requires knowledge the latest value ${𝒚}_{n+1}$. Approximate this by a predictor ${\stackrel{\sim }{𝒚}}_{n+1}$ obtained by an Adams-Bashforth of fourth order where needed, e.g., $𝒇\left({𝒚}_{n+1}\right)\cong 𝒇\left({\stackrel{\sim }{𝒚}}_{n+1}\right)$.