The Project
Numerical analysis and simulation of geometric wave equations.
Winner
Snorre Christiansen
University of Oslo
Centre of Mathematics for Applications
Oslo, Norway
www.cma.uio.no
Norwegian Snorre Christiansen, 30 years old, graduated from the Ecole Polytechnique, Paris, in 1997 with an Ingenieur Diplome and obtained his Ph.D in 2002. He is a popular requested speaker and lecturer at scientific conferences and seminars, participating in over 30 such events during the past five years. Christiansen is completing his post doctorate at the Centre of Mathematics for Applications, University of Oslo, Norway.
Award
€1,199,196
Project Description
Numerical simulations have grown to become a significant component of almost all scientific disciplines, playing the role of physical experiments in many circumstances. Yet for many mathematical models and in particular in the new field of gravitational wave astronomy, current algorithms are unsatisfactory, probably because of a lack of understanding of the relationship between continuum models and discrete models.
In this project, we will address the problem of designing good discretizations of geometric wave equations, including scalar waves, (classical) Yang-Mills equations and Einstein's equations of general relativity. With various degrees of non-linearity and difficulty such equations nevertheless have in common a rich algebraic structure, inherited from differential geometry and including invariance under certain groups of transformations.
We will develop tools enabling the numerical analysis of discretization of such equations, drawing on the experience of finite element discretizations of partial differential equations and structure preserving algorithms for ordinary differential equations. While finite element theory has been mainly developed in the framework of Sobolev spaces, we aim to adapt some of the more elaborate methods developed for non-linear equations in the calculus of variations. Mimicking the algebraic structure of the continuum model in the construction of algorithms, is a major trend in contemporary numerical analysis which has yet to realize its full potential for partial differential equations where the interplay with topological properties becomes crucial. Geometric wave equations provide a field of choice to bridge this gap on physically relevant systems.
