Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation type. I: RK- methods
For implicit Runge-Kutta (RK) methods applied to singularly perturbed systems of ordinary differential equations it is shown that the resulting discrete systems preserve the geometric properties of the underlying differential equation. In particular, if the perturbation parameter \(\varepsilon\) is small, under the usual assumptions, the singularly perturbed system has a highly attractive invariant manifold \(M_ \varepsilon\). For stepsize \(h\) fixed and \(\varepsilon\) small enough, the discrete system given by an implicit RK-method admits an attractive invariant manifold \(M_{h,\varepsilon}\) close to \(M_ \varepsilon\). This invariant manifold result is used to derive sharp bounds on the global error of RK-solutions.
scientific article; zbMATH DE number 770158