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Stiffness and stability

A stiff problem has widely different time scales, so an explicit method like Euler needs tiny steps to stay stable, even though the solution otherwise changes slowly. The stability limit for explicit Euler on $y' = \lambda y$ is $h \leq 2/|\lambda|$ .

|1 + h\lambda| \leq 1

stability requirement for explicit Euler on

y' = \lambda y

Symbols

$\lambda$	the growth rate in the equation (negative for a decaying solution)
$h$	step size

Example

$y' = -50y$ : stability requires $h \leq 2/50 = 0.04$ .

A larger step gives a solution that oscillates and grows unrealistically.

For stiff problems, switch to an implicit method (e.g. backward Euler) instead of just shortening

h

Practise differential equations and stability for free →

← Runge–Kutta (RK4) · System of first-order equations →

Part of Numerical Methods: Differential equations and stability.