A general method for obtaining unconventional and nonstandard difference schemes

In recent years it has been shown that some unconventional or nonstandard finite difference schemes have very good properties. Some are exact schemes, others give dynamically correct approximations for wide parameter ranges and arbitrary initial values even in the case of blow-up, or they are symplectic on a non-canonical Hamiltonian system. Thus it is highly desirable to know a general method for generating such schemes for any given differential equation, without a priori knowledge of the solution. We show that the ''linearized trapezoidal rule'' is such a method: some nonstandard schemes found in the literature are generated by it, others are approximated in a surprizing way. Moreover, the method is nearly a standard method: practitioners in computational plasma physics and fluid dynamics have been using it for years, and it belongs to the family of Rosenbrock methods. (orig.)20 refs.Available from TIB Hannover: RA 71(6/334) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman