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## An ODE is solved numerically three different ways, each using a time step of t=0.1  (Euler’s method, the midpoint method and a 4th order Runge Kutta method), and then again using t=0.1 .  The solution is NOT shown!  (neither is the problem!)  However, the errors in the solution (measured at t = 10) using that time step are shown in the table below:   Euler Error Midpoint Error Runge-Kutta Error 1 1.3 0.20 0.05 0.1 0.13 0.0020 5 x 10^6           For each method, write an approximate function which describes how the error varies with time-step .

An ODE is solved numerically three different ways, each using a time step of t=0.1  (Euler’s method, the midpoint method and a 4th order Runge Kutta method), and then again using t=0.1 .  The solution is NOT shown!  (neither is the problem!)  However, the errors in the solution (measured at t = 10) using that time step are shown in the table below:

 Euler Error Midpoint Error Runge-Kutta Error 1 1.3 0.20 0.05 0.1 0.13 0.0020 5 x 10^6

For each method, write an approximate function which describes how the error varies with time-step .

E_euler = …

E_mp = …

E_rk4 = …

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