Dahlquist, Liniger and Nevanlinna devised a family of one-leg two-step methods (DLN) that is second order, A- and G- stable for arbitrary, non-uniform time steps. The DLN method thus has strong potential for use in adaptive codes, but its adaptive step size selection is little explored. This report develops two approaches for the efficient local error estimation in the DLN method, and tests their use in a standard adaptivity framework. Many methods of error estimation are possible; herein we focus on two complementary estimators which involve minimal extra storage and computations. First we evaluate the local truncation error of the DLN method by Milne’s device, using the difference between the solution of the DLN method and the solution of a variable-step, explicit, second-order Adams-Bashforth-like method. Second, we use a recent refactorization of the DLN method, which eases implementation of DLN in legacy codes, to obtain an effective error estimation at no extra cost. We perform a number of numerical tests, comparing the two time adaptive DLN algorithms with some standard numerical ODE packages. Our tests indicate that the adaptive DLN method, with error estimated by Milne’s device, is an efficient and reliable method, even for stiff and unstable problems.

Sarcocystis neurona is considered the major etiologic agent of equine protozoal myeloencephalitis (EPM), a neurological disease in horses. Raccoon ( Procyon lotor ) is considered the most important intermediate host in the life cycle of S. neurona in the United States; S. neurona sarcocysts do mature in raccoon muscles, and raccoons also develop clinical signs simulating EPM. The focus of this study was to determine if sarcocysts would develop in raccoons experimentally inoculated with different host-derived strains of in vitro-cultivated S. neurona merozoites. Four raccoons were inoculated with strains derived from a raccoon, a sea otter, a cat, and a horse. Raccoon tissues were fed to laboratory-raised opossums ( Didelphis virginiana ), the definitive host of S. neurona . Intestinal scraping revealed sporocysts in opossums who received muscle tissue from raccoons inoculated with the raccoon-derived or the sea otter-derived isolates. These results demonstrate that sarcocysts can mature in raccoons inoculated with in vitro-derived S. neurona merozoites. In contrast, the horse and cat-derived isolates did not produce microscopically or biologically detected sarcocysts. Immunoblot analysis revealed both antigenic and antibody differences when testing the inoculated raccoons. Immunohistochemical staining indicated differences in staining between the merozoite and sarcocyst stages. The successful infections achieved in this study indicates that the life cycle can be manipulated in the laboratory without affecting subsequent stage development, thereby allowing further purification of strains and artificial maintenance of the life cycle.