# I have discovered that in solving ordinary differential equations things can get very easy if we use micro-intervals (intervals divided by 10 000 or more) and very rough methods like Euler's , to get the same accuracy as when using complex methods.

58 views (last 30 days)
Carlos Steinstrasser on 15 Jul 2019
Edited: Jan on 22 Jul 2019
ODEs of any order with initial conditions can be solved easely "using micro-intervals (intervals divide by more than a 1000) associated with Euler's method - definition of derivative -, can lead to great accuracy. I have a batch of scripts that can prove that. (study1, study2 ......). The simplicity is the great asset. Is this known ?

Carlos Steinstrasser on 19 Jul 2019
@Bjorn: No I didn't, because I need to understand completely the model and what I'm looking for. Maxwell equations are not "my thing". Otherwise, the system of first order equations are very basic in appearance and Euler certainly can be used. See the Jim Riggs's explanation above.
Bruno Luong on 19 Jul 2019
"I'd just like to raise a flag that what was a great idea in the past may have to be reconsidered because of the new computers. "
New computer? The double precision floating point 64 bits exists almost at the born of the computers, at least already in 1953 with FORTRAN, more than 50 years ago.
Not sure about the history, but it's not a surprise me that success of the moon landing in 1969 is greatly due to RK methods.
And the study of errors of RK methods mainly focus on the high order approximation of the solutions and little related to the truncation AFAIK.
Bjorn Gustavsson on 19 Jul 2019
OK, I can help explain the equation. As you've understood it is a simple system of first order equations. It is, however, not at al related to Maxwell's equations, it is the Newtonian equation of motion:
Further, you are correct in that an explicit Euler-method can be used. That is not the question here. The question is whether Euler-methods are suitable for this problem. For these kind of clean physical problems we know that there are constants of motion that should be conserved. Here the total energy is fixed - in this case only the kinetic energy of the single particle, i.e. should be constant, the particle should neither gain nor loose energy, mening the speed should be constant. This requirement is something that is in no way a given for general-purpose ODE-methods, Euler, R-K or other schemes, but a complementary abd completely different physcis-based requirement. For this ODE-schemes like the Størmer-Verlet or Boris-mover are preferable. So this question falls outside from Jim Rigg's eplanation above.
The equation as implemented above are in normalized units such that the gyro-period is , the first two components are for the time-derivative of particle position (x and y) and the two last components are the corresponding accellerations.