Euler methods of solving ODES Ordinary Differential Equations may well be the simplest solution methods, but are prone to errors during evaluation.

A number of steps are used to advance a parameter through time (or space, depending on the equation). For example:

d/dt(N)=-k.N

(rate of change of number proportional to the population).

The Euler method is:

  time=0
 loop (until fed up) {
   N=N-k.N*dt
   time=time+dt
 }

Or in tcl

 proc test {} {
   set mass 1
   set dt 0.001 ;# try shortening dt to see numerical inaccuracy
 # use single equation Euler scheme- log decay of radioactivity, dN/dt=-n solution: N(t)=exp(-t)
  set tpr 0.1
  set nextpr $tpr
  for {set time 0} {[expr $time<=2] } { set time [expr $time+$dt]} {
    if {$time>=$nextpr} {
        puts "At time $time mass is $mass"
          set nextpr [expr $nextpr+$tpr]
        }
    set mass [expr $mass-1.0*$mass*$dt] ;# dmass/dt = -mass - mass=m0(1-exp(time/k))
  }
 }
 test

The solution at time=1 is 0.3486784401; the theoretical value is 0.367879441171 and Runge-Kutta with the same problem and time step gives 0.367885238126.

For Euler achieve similar accuracy to RK needs a time step of 0.001 (and the solution is only 0.36769542477, 3 digits). Thus rk uses 4 evaluations, Euler needs 1000 evaluations of the driving function. The winner is R-K!

AM Note that R-K implicitly assumes very smooth solutions to achieve the accuracy. This is not always the case. Also R-K loses usefulness when it comes to partial differential equations.

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