Plenary Speaker at the XXXIX DYNAMICS DAYS EUROPE https://dyndays.uni-rostock.de/

The study of mathematical models whose dynamics go to infinity -blow up- in finite time is a topic of widespread interest in many areas of mathematics and physics. We address a numerical method that, upon suitable transformations allows the system to ``go beyond infinity" to the other side, with the solution becoming again not-singular and the numerical computations continuing normally. In Ordinary Differential Equations (ODE) the ``crossing" of infinity can happen instantaneously; In Partial Differential Equations (PDEs) the crossing of infinity persists for a finite time, and it is also mobile in space necessitating the introduction of computational buffer zones in which an appropriate singular transformation is continuously (locally) detected and performed. The proposed numerical approach could set the stage for a systematic numerical analysis of blowing up dynamics bypassing infinity in a broader range of evolution biologically and physically inspired examples.