Stiff Equation - Etymology

Etymology

The origin of the term 'stiffness' seems to be somewhat of a mystery. According to J. O. Hirschfelder, the term 'stiff' is used because such systems correspond to tight coupling between the driver and driven in servomechanisms. According to Richard. L. Burden and J. Douglas Faires,

Significant difficulties can occur when standard numerical techniques are applied to approximate the solution of a differential equation when the exact solution contains terms of the form eλt, where λ is a complex number with negative real part. ... Problems involving rapidly decaying transient solutions occur naturally in a wide variety of applications, including the study of spring and damping systems, the analysis of control systems, and problems in chemical kinetics. These are all examples of a class of problems called stiff (mathematical stiffness) systems of differential equations, due to their application in analyzing the motion of spring and mass systems having large spring constants (physical stiffness).

For example, the initial value problem

$m \ddot x + c \dot x + k x = 0, \qquad x(0) = x_0, \qquad \dot x(0) = 0, \qquad \qquad (10)$

with m = 1, c = 1001, k = 1000, can be written in the form (5) with n = 2 and

$\bold A = \left( \begin{array}{rr} 0 & 1 \\ -1000 & -1001 \end{array} \right), \qquad \qquad \qquad \qquad \qquad \qquad \quad (11)$

$\bold f(t) = \left( \begin{array}{c} 0 \\ 0 \end{array} \right), \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (12)$

$\bold x(0) = \left( \begin{array}{c} x_0 \\ 0 \end{array} \right), \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (13)$

and has eigenvalues . Both eigenvalues have negative real part and the stiffness ratio is

which is fairly large. System (10) then certainly satisfies statements 1 and 3. Here the spring constant k is large and the damping constant c is even larger. (Note that 'large' is a vague, subjective term, but the larger the above quantities are, the more pronounced will be the effect of stiffness.) The exact solution to (10) is

$x(t) = x_0 \left( - \frac{1}{999} e^{-1000 t} + \frac{1000}{999} e^{-t} \right) \approx x_0 e^{-t}. \qquad \qquad \qquad (15)$

Note that (15) behaves quite nearly as a simple exponential x₀e−t, but the presence of the e−1000t term, even with a small coefficient is enough to make the numerical computation very sensitive to step size. Stable integration of (10) requires a very small step size until well into the smooth part of the solution curve, resulting in an error much smaller than required for accuracy. Thus the system also satisfies statement 2 and Lambert's definition.

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