Symposium: Numerical Methods for Stiff Problems

摘要

The goal of the minisymposium is to consider some aspects related to the numerical treatment of stiff problems, with particular application to partial differential equations. Stiff problems arise in many contexts, and there is a vast literature of implicit schemes that are able to capture the solution of systems of differential equations, without fully resolve the smallest time scale of the problem. When a system comes from the discretization of an evolutionary partial differential equation, new interesting probelms arise. For exmple, the stiffnes of the problem may be space dependent, and standard discretization based on a method of line appeoach may become inefficient. In many cases, the stiffness may be expressed by a small parameter. Formal asymptotic of the original equation in the stiffness parameter gives an indication of the behavior of the equation for small values of such parameter. It is important that the numerical schemes are able to capture the asymptotic behavior of the solution. This is the case, for example, of singly perturbed problems. Two particular problems coming from partial differential equations will be considered. The first one concerns detonation problems, in which the challenge is to construct underresolved schemes (in space and time) which are able to capture the correct behavior fo the solution even for small relaxation time. The second problem concerns the construction of IMEX Runge-Kutta schemes applied to hyperbolic systems with diffusive relaxation, which are able to capture the diffusive limit of the system without the classical stability restriction of explicit schemes for the diffusion equation, restriction which is typical of the most recent approaches in the literature for this kind of problems.