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To the best of my knowledge, stiffness of ordinary differential equations is difficult to capture but can be roughly described as problems where explicit methods don't work while implicit ones do. Alternatively, the stiffness ratio of the Jacobian of the dynamics (i.e. the ratio of its largest to smallest eigenvalues) is supposed to be a quantitative indicator of a problem's stiffness. The advantage of implicit over explicit methods seems to be that they remain stable for larger time steps, even in the face of stiffness.

Does that mean that stiffness and instability are fundamentally linked? As stability is a property of the solver and stiffness a property of the problem, I would think that would not be the case. Are there any problems that are not stiff (e.g. have a small stiffness ratio) but explicit methods are still unstable on them? On the flip side, are there any problems with large stiffness that can be solved with explicit methods?

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2 Answers 2

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There are non-stiff problems which are unconditionally unstable with some explicit methods, and conversely there are stiff problems which can be stable with explicit methods. Consider the oscillating problem:

$$ \partial_t a = -b\\ \partial_t b = a $$ This has the eigenvalues $\pm i$, which lies outside the stability region of forwards Euler for any $\Delta t > 0$. Thus attempting to use forward Euler will be unconditionally unstable, despite this problem not being stiff.

However, there do exist explicit methods which can solve this problem. For example, the classical RK4 method includes some portion of the pure imaginary axis in its stability region.

On the other hand, we can consider the "very stiff" problem: $$ \partial_t a = b\\ \partial_t b = -a - 1000 b $$ which has the eigenvalues $\lambda \approx [-999.999, -0.001]$. It is possible to use forward Euler to solve this problem, however you are simply restricted to a "small" timestep dictated by the larger eigenvalue.

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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.

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