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 = ...
We know that the differential equation is solved using three different methods at two different time steps, t = 0.1, t = 1.
We are given the value of error obtained in solutions, from different methods at both the time steps.
The objective here is to form a relation between the time step used and the error obtained, for each method.
The data is given as follows:
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