Problem-based numerical methods for some local and non-local models

The numerical solution of real-life models cannot disregard the behavior of the analytical solution and/or the preservation of its special properties, such as for example the periodicity, the stiffness, the (lack of) smoothness in some intervals. In this talk it will be illustrated how this approach led to effective numerical methods, both for differential and integral models. Many practical problems in science and engineering are modeled by large systems of ordinary differential equations which arise from discretization in space of partial differential equations. For such systems there are often natural splittings of the right hand sides of the differential systems into two parts, one of which is non-stiff or mildly stiff, and suitable for explicit time integration, and the other part is stiff, and suitable for implicit time integration. Thus, here it is proposed an implicit-explicit (IMEX) scheme based on GLM methods, which has the advantage of preserving the order of the composing methods and to have enough free parameters to optimize the stability properties. On the side of non local models, it will be illustrated the numerical discretization of Volterra integral equations (VIEs) with periodic solution and of fractional differential equations (FDEs), both suitable to model problems with memory. VIEs with periodic solution can represent periodic phenomena with memory, like the spread of seasonal diseases. Classical methods are able to follow the oscillations of the solution at a high computational cost, while the exponentiallyfitted methods that we propose can considerably reduce this cost by exploiting the knowledge of an estimation of the frequency. FDEs can model the anomalous kinetics of some processes in physics, chemistry, pharmacokinetics. It will be illustrated a spectral collocation method, which takes into account the non-local nature of the equation, with a function basis suitably chosen to reproduce the behavior of the analytical solution. This presentation is based on the research work carried out with M. Bras (AGH Univ., Poland), K. Burrage (Oxford Univ.), D. Conte (Univ. of Salerno), R. D’Ambrosio (Univ. of L’Aquila), L.Gr. Ixaru (“Horia Hulubei” Nat.Inst. Physics & Nuclear Eng., Romania), Z. Jackiewicz (Arizona State Univ.), B. Paternoster (Univ. of Salerno), A. Sandu (Virginia Polytechnic Inst. & State Univ.), G.Santomauro (ENEA) and H. Zhang (Argonne Nat. Lab.)