SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Plenary Lecture


On solving highly oscillatory ODE problems

A. Murua

Abstract
In this talk, we are interested in studying the solutions of highly oscillatory systems of ordinary differential equations and constructing and analyzing numerical integrators for such problems. We present the approach developed in [1, 2, 3, 4, 5], where standard combinatorial-algebraic tools for analysing numerical integrators for non-oscillatory ODEs are adapted to the highly oscillatory context. We explore in which extent our approach for analysing the solutions of highly oscillatory problems can by used to construct actual numerical integrators tailored for such problems by treating in some detail several examples that fit in our framework.

Bibliography
[1] P. Chartier, A. Murua and J.M. Sanz-Serna, Higher-Order averaging, formal series and numerical integration I: B-series, Found. Comput. Math. 10, 695-727 (2010).
[2] P. Chartier, A. Murua and J.M. Sanz-Serna, Higher-Order averaging, formal series and numerical integration II: the quasi-periodic case, Found. Comput. Math. 12, 471-508 (2012).
[3] P. Chartier, A. Murua and J.M. Sanz-Serna, A formal series approach to averaging: exponentially small error estimates, DCDS A 32, 3009-3027 (2012).
[4] Ph. Chartier, A. Murua and J.M. Sanz-Serna, Higher-order averaging, formal series and numerical integration III: error bounds, submitted.
[5] M.P. Calvo, Ph. Chartier, A. Murua and J.M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math. 61, 1077-1095 (2011).

Organized by         Universidad de Valladolid     IMUVA