StiffnessTest Method Option for NDSolve

The methods DoubleStep, Extrapolation, and ExplicitRungeKutta have the option StiffnessTest, which can be used to identify whether the method applied with the specified AccuracyGoal and PrecisionGoal tolerances to a given problem is stiff.

A convenient way of detecting stiffness is to directly estimate the dominant eigenvalue of the Jacobian J of the problem (see [S84] and [R87]).

Such an estimate is often available as a by-product of the numerical integration and so it is reasonably inexpensive.

If v denotes an approximation to the eigenvector corresponding to dominant eigenvalue of the Jacobian, with v sufficiently small, then by the mean value theorem a good approximation to the leading eigenvalue is:

Richardson's extrapolation provides a sequence of refinements that yield a quantity of this form as do certain explicit Runge-Kutta methods.

Let LSB denote the linear stability boundary—the intersection of the linear stability region with the negative real axis.

The product gives an estimate that can be compared to the linear stability boundary of a method in order to detect stiffness.

The method option StiffnessTest itself accepts a number of options that are now illustrated.

The default values for the stiffness test implement a weak form of (1) where the test incorporates a safety factor and is allowed to fail a specified number of times.

The reason for this is that some problems can be only mildly stiff in a certain region and an explicit integration method may still be efficient.