Explicit vs. Implicit Algorithms

Explicit vs. Implicit Algorithms

Explicit: predicted y(t+h) depends only on y(t),y'(t) at earlier times. Cheaper time steps, but poor stability may strictly limit h.

Implicit: predicted y(t+h) depends on y'(t+h) or other future-time information. Expensive time steps, but much more stable.

Ideal Pendulum:

d /dt =
d /dt = -(g/L) sin

Explicit (Forward Euler):

(t+h) = (t) + h (t)
(t+h) = (t) - (hg/L) sin (t)

Implicit (Backward Euler):

(t+h) = (t) + h (t+h)
(t+h) = (t) - (hg/L) sin (t+h)

Must solve

(t+h) = (t) + h (t) - (h2g/L) sin (t+h)

for (t+h)

Semi-Implicit: use a single Newton-Raphson approximate the root for an implicit method.

Predictor-Corrector: Use an explicit method to get an initial guess for the root of an implicit method.