[Math] Stiff differential equation where Runge-Kutta $4$th order method can be broken

numerical methodsordinary differential equations

Is there a stiff differential equation that cannot be solved by the Runge-Kutta 4th order method, but which has an analytical solution for testing?

Best Answer

How about "impractical"? Here's an example from Numerical Recipes in C, Second Ed., Sec. 16.6:

$$u'=998 u+1998 v$$

$$v'=-999 u -1999 v$$

where $u(0) = 1$ and $v(0) = 0$.

Then

$$u(x) = 2 e^{-x}- e^{-1000 x}$$

$$v(x) = -e^{-x} + e^{-1000 x}$$

Runge-Kutta 4th order would require a stepsize of less than $0.001$ for any accuracy, which is obviously aggravating for the one of the solutions which could be acquired with a far coarser step.

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