Stiff equation

Motivating example

Consider the initial value problem

The exact solution (shown in cyan) is

We seek a numerical solution that exhibits the same behavior.

The figure (right) illustrates the numerical issues for various numerical integrators applied on the equation.

One of the most prominent examples of the stiff ODEs is a system that describes the chemical reaction of Robertson:

If one treats this system on a short interval, for example, t ∈ [ 0 , 40 ] there is no problem in numerical integration. However, if the interval is very large (10¹¹ say), then many standard codes fail to integrate it correctly.

Additional examples are the sets of ODEs resulting from the temporal integration of large chemical reaction mechanisms. Here, the stiffness arises from the coexistence of very slow and very fast reactions. To solve them, the software packages KPP and Autochem can be used.

Stiffness ratio

Consider the linear constant coefficient inhomogeneous system

where y , f ∈ R n and A is a constant n × n matrix with eigenvalues λ t ∈ C , t = 1 , 2 , … , n (assumed distinct) and corresponding eigenvectors c t ∈ C n , t = 1 , 2 , … , n . The general solution of (5) takes the form

where the κ_t are arbitrary constants and g ( x ) is a particular integral. Now let us suppose that

which implies that each of the terms exp ⁡ ( λ t x ) c t → 0 as x → ∞ , so that the solution y ( x ) approaches g ( x ) asymptotically as x → ∞ ; the term exp ⁡ ( λ t x ) c t will decay monotonically if λ_t is real and sinusoidally if λ_t is complex. Interpreting x to be time (as it often is in physical problems) it is appropriate to call Σ t = 1 n κ t exp ⁡ ( λ t x ) c t the transient solution and g ( x ) the steady-state solution. If | R e ( λ t ) | is large, then the corresponding term κ t exp ⁡ ( λ t x ) c t will decay quickly as x increases and is thus called a fast transient; if | R e ( λ t ) | is small, the corresponding term κ t exp ⁡ ( λ t x ) c t decays slowly and is called a slow transient. Let λ ¯ , λ _ ∈ { λ t , t = 1 , 2 , … , n } be defined by

so that κ t exp ⁡ ( λ ¯ x ) c t is the fastest transient and κ t exp ⁡ ( λ _ x ) c t the slowest. We now define the stiffness ratio as

Characterization of stiffness

In this section we consider various aspects of the phenomenon of stiffness. 'Phenomenon' is probably a more appropriate word than 'property', since the latter rather implies that stiffness can be defined in precise mathematical terms; it turns out not to be possible to do this in a satisfactory manner, even for the restricted class of linear constant coefficient systems. We shall also see several qualitative statements that can be (and mostly have been) made in an attempt to encapsulate the notion of stiffness, and state what is probably the most satisfactory of these as a 'definition' of stiffness.

J. D. Lambert defines stiffness as follows:

If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a steplength which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval.

There are other characteristics which are exhibited by many examples of stiff problems, but for each there are counterexamples, so these characteristics do not make good definitions of stiffness. Nonetheless, definitions based upon these characteristics are in common use by some authors and are good clues as to the presence of stiffness. Lambert refers to these as 'statements' rather than definitions, for the aforementioned reasons. A few of these are:

1. A linear constant coefficient system is stiff if all of its eigenvalues have negative real part and the stiffness ratio is large.
2. Stiffness occurs when stability requirements, rather than those of accuracy, constrain the steplength.
3. Stiffness occurs when some components of the solution decay much more rapidly than others.

Etymology

The origin of the term 'stiffness' seems to be somewhat of a mystery. According to Joseph Oakland 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 e^λt, where λ is a complex number with negative real part.

...

For example, the initial value problem

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

and has eigenvalues λ ¯ = − 1000 , λ _ = − 1 . 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

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.

A-stability

The behaviour of numerical methods on stiff problems can be analyzed by applying these methods to the test equation y' = ky subject to the initial condition y(0) = 1 with k ∈ C . The solution of this equation is y (t) = e^kt. This solution approaches zero as t → ∞ when R e ( k ) < 0. If the numerical method also exhibits this behaviour (for a fixed step size), then the method is said to be A-stable. (Note that a numerical method that is L-stable (see below) has the stronger property that the solution approaches zero in a single step as the step size goes to infinity.) A-stable methods do not exhibit the instability problems as described in the motivating example.

Runge–Kutta methods

Runge–Kutta methods applied to the test equation y ′ = k ⋅ y take the form y n + 1 = ϕ ( h k ) ⋅ y n , and, by induction, y n = ( ϕ ( h k ) ) n ⋅ y 0 . The function ϕ is called the stability function. Thus, the condition that y n → 0 as n → ∞ is equivalent to | ϕ ( h k ) | < 1 . This motivates the definition of the region of absolute stability (sometimes referred to simply as stability region), which is the set { z ∈ C | | ϕ ( z ) | < 1 } . The method is A-stable if the region of absolute stability contains the set { z ∈ C | R e ( z ) < 0 } , that is, the left half plane.

Example: The Euler methods

Consider the Euler methods above. The explicit Euler method applied to the test equation y ′ = k ⋅ y is

Hence, y n = ( 1 + h k ) n ⋅ y 0 with ϕ ( z ) = 1 + z . The region of absolute stability for this method is thus { z ∈ C | | 1 + z | < 1 } which is the disk depicted on the right. The Euler method is not A-stable.

The motivating example had k = − 15 . The value of z when taking step size h = 1 / 4 is z = − 15 ∗ 1 / 4 = − 3.75 , which is outside the stability region. Indeed, the numerical results do not converge to zero. However, with step size h = 1 / 8 , we have z = − 1.875 which is just inside the stability region and the numerical results converge to zero, albeit rather slowly.

Example: Trapezoidal method

Consider the trapezoidal method

when applied to the test equation y ′ = k ⋅ y , is

Solving for y n + 1 yields

Thus, the stability function is

and the region of absolute stability is

This region contains the left-half plane, so the trapezoidal method is A-stable. In fact, the stability region is identical to the left-half plane, and thus the numerical solution of y ′ = k ⋅ y converges to zero if and only if the exact solution does. Nevertheless, the trapezoidal method does not have perfect behavior: it does damp all decaying components, but rapidly decaying components are damped only very mildly, because ϕ ( z ) → 1 as z → − ∞ . This led to the concept of L-stability: a method is L-stable if it is A-stable and | ϕ ( z ) | → 0 as z → ∞ . The trapezoidal method is A-stable but not L-stable. The implicit Euler method is an example of an L-stable method.

General theory

The stability function of a Runge–Kutta method with coefficients A and b is given by

where e denotes the vector with ones. This is a rational function (one polynomial divided by another).

Explicit Runge–Kutta methods have a strictly lower triangular coefficient matrix A and thus, their stability function is a polynomial. It follows that explicit Runge–Kutta methods cannot be A-stable.

The stability function of implicit Runge–Kutta methods is often analyzed using order stars. The order star for a method with stability function ϕ is defined to be the set { z ∈ C | | ϕ ( z ) | > | e z | } . A method is A-stable if and only if its stability function has no poles in the left-hand plane and its order star contains no purely imaginary numbers.

Multistep methods

Linear multistep methods have the form

Applied to the test equation, they become

which can be simplified to

where z = hk. This is a linear recurrence relation. The method is A-stable if all solutions {y_n} of the recurrence relation converge to zero when Re z < 0. The characteristic polynomial is

All solutions converge to zero for a given value of z if all solutions w of Φ(z,w) = 0 lie in the unit circle.

The region of absolute stability for a multistep method of the above form is then the set of all z ∈ C for which all w such that Φ(z,w) = 0 satisfy |w| < 1. Again, if this set contains the left-half plane, the multi-step method is said to be A-stable.

Example: The second-order Adams–Bashforth method

Let us determine the region of absolute stability for the two-step Adams–Bashforth method

The characteristic polynomial is

which has roots

thus the region of absolute stability is

This region is shown on the right. It does not include all the left half-plane (in fact it only includes the real axis between z = −1 and z = 0) so the Adams–Bashforth method is not A-stable.

General theory

Explicit multistep methods can never be A-stable, just like explicit Runge–Kutta methods. Implicit multistep methods can only be A-stable if their order is at most 2. The latter result is known as the second Dahlquist barrier; it restricts the usefulness of linear multistep methods for stiff equations. An example of a second-order A-stable method is the trapezoidal rule mentioned above, which can also be considered as a linear multistep method.