Several extreme natural disasters in recent years - to mention the 2004 Sumatra-Andaman and 2011 Tohoku Tsunami events, 2005 Hurricane Katrina, 2013 Taifun Haiyan, 2010 Eyjafjallajökull Volcano eruption - reminded the world of the vulnerability of modern societies and highly populated areas. In order to save life and property, early warning and preparedness is essential, which in turn requires fast and accurate simulation results for forecasting and scenario simulations. This presentation discusses requirements in such operations from a mathematical and computational point of view. One major concern is the multi-scale character of many such problem settings. Solution approaches to the requirements of multi-scale representation, accuracy, efficiency and visualization are presented. We try to extract overarching computational strategies for disaster mitigation.