2010-03-23
10:15 at HCI J 498

Structure preserving discretizations for GENERIC evolution problems

Ignacio Romero

Technical University of Madrid

The development of numerical methods which preserve the mathematical structure of the equations whose solution they are meant to approximate has guided a large part of the efforts of numerical analysts during the last two decades. In Mechanics, most of the advances have taken place in the numerical solution of Hamiltonian problems, whose well-known mathematical structure has guided the algorithmic design. In contrast, fewer results have been obtained for general evolution equations since no formalism exists that can encompass all of them. In those cases where a structure preserving discretization has been proposed, for example, for a dissipative problem, the methods were applicable only to those specific equations of evolution and thus have limited impact. The GENERIC formalism for nonequilibrium thermodynamics offers a fairly general framework which allows the unified description of many problems of evolution with applications to physics, chemistry, and engineering. This generality makes it a perfect candidate for guiding the design of structure preserving algorithms, and we have recently proposed discretizations that strictly preserve the most fundamental ingredients of GENERIC: the separation of flows derived from the reversible and irreversible phenomena and the satisfaction of the two laws of thermodynamics. In addition, the methods we have tested are extremely stable. In the talk, I would review structure preserving methods for Hamiltonian problems, describe the new GENERIC preserving algorithms, and show numerical examples for non-trivial infinite dimensional evolution problems.