Considering the simple dynamical system as follows

where and are sparse matrices, is a input and is incident matrix which poses in the system.

In my thought, the explicit methods work if there is not on the left hand side,

e.g. solve it by forward Euler method and ignore the input for simplicity (), then

, where is the discretized time step.

However, when this is not true, the explicit method still need to solve linear system, because

or say it requires inversion of matrix

Any comments?

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[Add by author] one good thing about explicit method is that the factorized matrix does not contain the step size, which makes it good for stepping tuning, where the numerical stability poses a hinder for even mild stiff case.

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