A conditionally P-stable fourth-order exponential-fitting method for y″=f(x,y)

摘要

We search for the modification of the stability properties of a P-stable multistep algorithm with an order greater than two when this is reformulated on the basis of the exponential fitting. We effectively construct the exponential-fitting version of the fourth-order two-step algorithm of Chawla (BIT 21 (1981) 190–193). In order to characterize the stability properties of the new version, we first introduce the concept of the conditional P-stability of a family of exponential-fitting methods and the parameter θmax to be associated to a conditionally P-stable family. We then show that this version is conditionally P-stable indeed, with θmax=3.4. This means that the method can be used for stiff problems provided some typically nonsevere restrictions are imposed on the stepsize h.