Numerical intergration of perturbed linear systems
The paper deals with numerical integration of the perturbed system of differential equations of the form \[ x'=Ax+\varepsilon g(x,t),\qquad x(0)=x_0, \tag{1} \] where \(x:{\mathbb R}\to {\mathbb R}^m, g:{\mathbb R}\times {\mathbb R}\to {\mathbb R}^m\) and \(A\) is a constant \(m\times m\) matrix. The authors suggest new explicit and implicit multistep algorithms for the integration of (1) based on the method of \textit{G. Scheifele} [Z. Angew. Math. Phys. 22, 186-210 (1971; Zbl 0221.56135)]. In order to avoid difficult preliminary computations of coefficients required for the implementation of the Scheifele method, the latter is transformed into a multistep scheme with the good properties of the original method preserved. Some numerical examples are presented to illustrate the advantage of a new method over the classical ones like Adams or Runge-Kutta methods.
scientific article; zbMATH DE number 1367739