I'm trying to prepare for my exam in ordinary differential equations and I've got this question. (Stiffness)

"fundamental difference between explicit and implicit schemes"

My own idea is that explicit method is based on previus calculations. I.e. Explicit Euler method. But how about Implicit Euler?

  • $\begingroup$ Do you know how implicit Euler, the implicit trapezoidal method or 2-stage Gauß work? What the equations are, what is trivial, what is non-trivial about them? What a predictor-corrector scheme is and how simplified Newton might give a slight advantage? $\endgroup$ – LutzL Jan 11 at 13:24

Implicit schemes are implicit, that is, you have to solve a non-linear equation or at least approximate the solution better than the discretization error.

Explicit methods are never A-stable, implicit methods can be A-stable, some even satisfy more specialized stability conditions. Stability is important for solving stiff equations, as it allows larger step sizes.


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