Recent problem-oriented approaches for the numerical solution of differential equations

This talk concerns recent advances in the numerical solution of evolutionary problems deriving from natural, chemical, physical and other phenomena, whose solutions are characterized by specific qualities such as positivity, oscillations and/or stiffness. We will describe the construction of strongly problem-oriented numerical methods, i.e. methods which are able to preserve the main qualitative features of the problem considered regardless of the choice of the discretization step-size. In particular, we will use temporal and/or spatial discretizations constructed by means of non-polynomial bases and non-standard finite differences, proving that in this way it is possible to preserve any positivity and oscillations frequency of the exact solution. For the solution of stiff differential equations arising, e.g., from the discretization in space of parabolic problems, we will describe the derivation of jacobian-dependent numerical methods with explicit structure and good stability properties. Moreover, we improve the accuracy and stability properties of explicit Runge-Kutta and peer methods by using TASE operators. Finally, we will present applications to models describing real phenomena, such as growth of vegetation in arid environments and pitting corrosion, also discussing the construction of appropriate software making use of Matlab built-in functions to improve the efficiency of computations.