I have solved the equation of motion of a spring mass damper system using RK method of 4th order (I wrote the algorithm, didn't use any ODE solver in MATLAB). I varied the time step from 0.5 to 10e-5 and observed that for time step less than 10e-3 the order of error in case of Euler's method and RK4 method is becoming same. Can anyone please help me out?
Thanks for your response Ali Mahdavi
. But I am not concerned about adaptive RK methods. I need to check for RK4 only. Even if I use the inbuild function runkut4 in MATLAB I am getting the similar trend of error.
What you see is quite typical, moving towards smaller step sizes h the accumulated errors from the summation of the small updates of the O(1/h) many steps start dominating the decreasing error O(h^4) from the method. The slope in the step-size error dependence changes from 4 to -1 (from right to left), your graph stops close to the minimum. You would see the V shape more clearly with the addition of smaller step sizes and more data points in-between. See
https://math.stackexchange.com/questions/3235194/truncation-error-with-growing-step-size
for an example. A more detailed statement to the error types I wrote in answer to
https://scicomp.stackexchange.com/questions/37425/how-error-scales-with-numerical-precision
For an example of how compensating the summation error as mentioned in the link can improve the integration see
https://math.stackexchange.com/questions/1210698/maximum-timestep-for-rk4
or
https://scicomp.stackexchange.com/questions/36619/floating-point-and-global-error
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