Explicit and Implicit Finite Difference Schemes

Explicit:

$$y_{n+1} = x_n + f(y_n)$$

Implicit:

$$y_{n+1} = x_n + f(y_{n+1})$$

• A finite difference scheme is said to be explicit when it can be computed forward in time using quantities from previous time steps

• We will associate explicit finite difference schemes with causal digital filters

• In implicit finite-difference schemes, the output of the time-update ($y_{n+1}$ above) depends on itself, so a causal recursive computation is not specified

• Implicit schemes are generally solved using
  • iterative methods (such as Newton's method) in nonlinear cases, and
  • matrix-inverse methods for linear problems

• Implicit schemes are typically used offline
  (not in real time)