Numerical Schemes as Approximations to Exp

Suppose we reach a system:

where and is a matrix.
(For example, by Semidiscretization)
The solution is given by:

If can be diagonalized as ,
then where
Computing such an eigenvalue decomposition can be expensive
We have a couple approximations

Euler

gives Euler method

Implicit Euler

gives Reverse-Euler Method

Trapezoidal

gives Trapezoidal rule (ODEs)