Three different integration schemes

[member 428835]

I'm solving a first order linear ODE system numerically three different ways: implicit Euler, explicit Euler, and RK4. Attached are the plots of the numerical solutions (line and stream of small closely-connected dots) and the exact solution (big dots). Also I plot the maximum error (right column).

My question is, why are some techniques better than others?

 

[DrClaude]

My question is, why are some techniques better than others?

A complete answer would necessitate writing a textbook on numerical methods. Therefore, my best recommendation is to read good resources on the subject.

In a nutshell, you have to remember that these numerical methods produce approximations to the actual solutions. For instance, RK4 is called that because it is accurate up to 4th order. Different methods will have different accuracies. Also, there is the question of propagation of error: how does the error in one step affect the error in a subsequent step? Some methods are unstable: it is impossible to keep the errors from accumulating to the point of producing incorrect results.

There is also a choice of what is being approximated out. For example, some integrators are symplectic, meaning that they will preserve the volume in phase space. Another method which is not symplectic might be of higher accuracy, but the price to pay in using it is a distortion in phase space. In quantum mechanics, it is often important to use methods that are unitary and will conserve the norm of the wave function.