implicit vs explicit
 

Hi,

I have written a little program to solve an ordinary dfferential equation. The user can decide whether an explicit or implicit integration method is used. Looking on the results, I cannot see any advantage of the implicit scheme.

What do you think about it?

The mathematical bakground was:

IMPLICIT: f(t+Dt) = f(t) + Dt* df/dt|_(t+Dt)

EXPLICIT: f(t+Dt) = f(t) + Dt* df/dt|_(t)

Re: implicit vs explicit
 

Try using f(t+Dt) = f(t) + 0.5*Dt*( df/dt|_(t+Dt) + df/dt|_(t) )

and vary the timestep comparing the results.

For an ode you'd should really look at explicit Runge-Kutta methods rather than your forward Euler step.

Re: implicit vs explicit
 

One way of looking at it is that if f(t) is mass say and therefore df/dt is mass flow, what happens if Dt*df/dt(t) is negative and higher in magnitude than f(t)?

In the explicit scheme you'll get negative mass (bad!). In the implicit scheme the iterated estimate of df/dt(t+Dt) will prevent the negative mass.

Explicit schemes assume that the grients at (t) are constant through the step. Implicit ones don't. Explicit schemes require small steps because of this.