An investigation of Runge-Kutta processes, and equivalence of scalar and vector cases
The application of Runge-Kutta methods of order m for systems: \(y'=f(x,y)\), \(y(x_ 0)=y_ 0\), generates a system of nonlinear algebraic equations whose number increases rapidly with m (reaching for \(m=14\), 3259 equations for the scalar case, 53272 for the vector case). The authors show how these numbers may be obtained for any m by counting certain types of tree graphs which correspond to the expression in the form of a tensor structure of the differential operators that appear in the process. They also give for fifth- and sixth-order formulae criteria under which both scalar and vector systems reduce to the same equivalent system. Some errors in previous concerning papers are indicated.
scientific article; zbMATH DE number 3880025