Adapted Numerical Modeling for Evolutionary Problems

The talk focuses on recent advances in the adapted numerical solution of evolutionary problems, mostly based on ordinary and partial differential equations. The performed adaptation is based on polynomial or non-polynomially fitting, wisely chosen in order to exploit the a priori knowledge of the qualitative behaviour of the solution, gaining advantages in terms of efficiency and accuracy with respect to classical schemes already known in literature. In the case of PDEs, the adaptation of the numerical scheme is carried out by finite differences computed in order to be exact on trigonometrical basis functions, coupled with an Implicit-Explicit (IMEX) time- integration or a peer numerical method. The coefficients of the resulting numerical scheme depend on unknown parameters to be properly estimated: such an estimate is performed by an efficient offline minimization of the leading term of the local truncation error. In the case of stiff ODEs, Jacobian-dependent vs Jacobian-free numerical discretizations are described, analyzed and compared. Moreover, when the problem is particularly stiff, a remedy to avoid the typical phenomenon of order reduction is presented, through collocation based multivalue numerical methods, providing a dense output of continuous uniform order on the overall integration interval. The effectiveness of above approaches is theoretically analyzed and also experimentally confirmed on a selection of test problems.