Besides stiffness problem, the explicit methods are still not better than implicit methods?

Considering the simple dynamical system as follows

{\bf C}{\bf \dot{x}}(t)={\bf G}{\bf }x(t)+{\bf B}{\bf u}(t)

where \bf{C} and \bf{G} are sparse matrices, {\bf u}(t) is a input and \bf{B} is incident matrix which poses {\bf u}(t) in the system.

In my thought, the explicit methods work if there is not \bf{C} on the left hand side,

{\bf \dot{x}}(t)={\bf G}{\bf x}(t)+{\bf B}{\bf u}(t)

e.g. solve it by forward Euler method and ignore the input for simplicity (\bf{\dot{x}}(t)={\bf G}{\bf x}(t))), then

{\bf x}(t+h)={\bf x}(t)+h{\bf G}{\bf x}(t), where h is the discretized time step.

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

{\bf C}{\bf x}(t+h)={\bf C}{\bf x}(t)+h{\bf G}{\bf x}(t) or say it requires inversion of matrix {\bf x}(t+h)={\bf x}(t)+h{\bf C}^{-1}{\bf G}{\bf x}(t)

Any comments?

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One thought on “Besides stiffness problem, the explicit methods are still not better than implicit methods?

  1. Hao Zhuang Post author

    [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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