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Laurent90
Laurent90
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I'd like to add a few complements to the accepted answer.

Some problems possessing some eigenvalues with positive real parts have "physically" unstable modes, which may actually be damped by many implicit methods (even some explicit ones which have some parts of their stability domain in the right half of the complex plane) if the time step is too large.

Finally, there is at least one class of stiff problems that can be solved with some specialised explicit methods competitively. It is the class of ODE systems whose eigenvalues $z$ lie very close to the negative real axis, such as obtained when semi-discretising in spa z the heat equation for instance. In.that case, the largest eigenvalue is $z\approx -(1/dx^2)$ with $dx$ your mesh size. Classical explicit methods have a stability domain whose extent is linear in $\Delta t$ and is proportional to the number of internal stages $s$ (for Runge-Kutta methods). Stabilized explicit methods, e.g. ROCK4, have been designed such that their stability domains are very close to the negative real axis and spread as far as possible towards $-\infty$. They typically have a dynamic number of internal stages, and their construction produces a stability domain whose extent is proportional to the square of the number of stages, $s^2$. This makes them highly efficient for many diffusion problems.

I'd like to add a few complements the accepted answer.

Some problems possessing some eigenvalues with positive real parts have "physically" unstable modes, which may actually be damped by many implicit methods (even some explicit ones which have some parts of their stability domain in the right half of the complex plane) if the time step is too large.

Finally, there is at least one class of stiff problems that can be solved with some specialised explicit methods competitively. It is the class of ODE systems whose eigenvalues $z$ lie very close to the negative real axis, such as obtained when semi-discretising in spa z the heat equation for instance. In.that case, the largest eigenvalue is $z\approx -(1/dx^2)$ with $dx$ your mesh size. Classical explicit methods have a stability domain whose extent is linear in $\Delta t$ and is proportional to the number of internal stages $s$ (for Runge-Kutta methods). Stabilized explicit methods, e.g. ROCK4, have been designed such that their stability domains are very close to the negative real axis and spread as far as possible towards $-\infty$. They typically have a dynamic number of internal stages, and their construction produces a stability domain whose extent is proportional to the square of the number of stages, $s^2$. This makes them highly efficient for many diffusion problems.

I'd like to add a few complements to the accepted answer.

Some problems possessing some eigenvalues with positive real parts have "physically" unstable modes, which may actually be damped by many implicit methods (even some explicit ones which have some parts of their stability domain in the right half of the complex plane) if the time step is too large.

Finally, there is at least one class of stiff problems that can be solved with some specialised explicit methods competitively. It is the class of ODE systems whose eigenvalues $z$ lie very close to the negative real axis, such as obtained when semi-discretising in spa z the heat equation for instance. In.that case, the largest eigenvalue is $z\approx -(1/dx^2)$ with $dx$ your mesh size. Classical explicit methods have a stability domain whose extent is linear in $\Delta t$ and is proportional to the number of internal stages $s$ (for Runge-Kutta methods). Stabilized explicit methods, e.g. ROCK4, have been designed such that their stability domains are very close to the negative real axis and spread as far as possible towards $-\infty$. They typically have a dynamic number of internal stages, and their construction produces a stability domain whose extent is proportional to the square of the number of stages, $s^2$. This makes them highly efficient for many diffusion problems.

Source Link
Laurent90
Laurent90
  • 2k
  • 5
  • 9

I'd like to add a few complements the accepted answer.

Some problems possessing some eigenvalues with positive real parts have "physically" unstable modes, which may actually be damped by many implicit methods (even some explicit ones which have some parts of their stability domain in the right half of the complex plane) if the time step is too large.

Finally, there is at least one class of stiff problems that can be solved with some specialised explicit methods competitively. It is the class of ODE systems whose eigenvalues $z$ lie very close to the negative real axis, such as obtained when semi-discretising in spa z the heat equation for instance. In.that case, the largest eigenvalue is $z\approx -(1/dx^2)$ with $dx$ your mesh size. Classical explicit methods have a stability domain whose extent is linear in $\Delta t$ and is proportional to the number of internal stages $s$ (for Runge-Kutta methods). Stabilized explicit methods, e.g. ROCK4, have been designed such that their stability domains are very close to the negative real axis and spread as far as possible towards $-\infty$. They typically have a dynamic number of internal stages, and their construction produces a stability domain whose extent is proportional to the square of the number of stages, $s^2$. This makes them highly efficient for many diffusion problems.