# Can Taylor methods be used effectively on stiff ODEs?

Cleve Moler has stated that "all numerical methods for stiff odes are implicit." However, I don't know whether this statement is a mathematical fact, or an simply an observation. Moreover, many numerical methods for ODEs assume that the only thing we can do with the right-hand side is point evaluation. If, for example, we can use automatic differentiation on the right-hand side of $$x' = f(x,t)$$, we are led into Taylor methods for ODEs.

But can Taylor methods be used effectively on stiff equations? If not, are there implicit Taylor methods which patch this problem up?

Note: There is always a problem about the definition of stiff. Will Cleve's definition work in this context?

A problem is stiff if the solution being sought varies slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results.

• Wasn’t one of the definitions of stiff that explicit methods need excessively long to (adequately) solve the problem? Jul 24 '19 at 7:57
• Yes, I'd agree with Wrzlprmft - it is more of a definition. If an explicit method is more efficient, the problem is not really stiff. Jul 24 '19 at 10:10
• Not all explicit methods. ROCK methods are a counter example to the idea that explicit methods are not able to be used on stiff equations. But effectively, yes, the idea is that the derivatives may be changing so that that extrapolating from a single point will be wildly unstable, so smoothing by using the future point (or other smoothing points) can be beneficial. Jul 24 '19 at 10:52