On the numerical preservation of qualitative properties of differential equations

We consider differential problems arising from real phenomena [1, 5], where some qualitative features and properties of the exact solution are a-priori known. Our aim is to develop numerical techniques that are able to preserve such features [4, 6], while also exhibiting excellent stability properties [2, 3]. In particular, we focus on stiff differential problems with positive and sometimes oscillating (with known frequency) solutions. We show numerical tests confirming the efficiency, stability and accuracy of the proposed numerical methods.