Janosch Rieger — Page 2

arXiv2013

Semi-implicit Euler schemes for ordinary differential inclusions

Two semi-implicit Euler schemes for differential inclusions are proposed and analyzed in depth. An error analysis shows that both semi-implicit schemes inherit favorable stability properties from the differential inclusion. Their performance is considerably better than that of the implicit Euler scheme, because instead of implicit inclusions only implicit equations have to be solved for computing their images. In addition, they are more robust with respect to spatial discretization than the implicit Euler scheme.

math.NA↗

arXiv2013

Robust boundary tracking for reachable sets of nonlinear differential inclusions

The Euler scheme is up to date the most important numerical method for ordinary differential inclusions, because the use of the available higher-order methods is prohibited by their enormous complexity after spatial discretization. Therefore, it makes sense to reassess the Euler scheme and optimize its performance. In the present paper, a considerable reduction of the computational cost is achieved by setting up a numerical method that computes the boundaries instead of the complete reachable sets of the fully discretized Euler scheme from lower-dimensional data only. Rigorous proofs for the propriety of this method are given, and numerical examples illustrate the gain of computational efficiency as well as the robustness of the scheme against changes of topology of the reachable sets.

math.NA↗