Graduate School in Nonlinear Science

Sponsored by The Danish Research Agency

MIDIT                              OFD                          CATS
Modelling, Nonlinear Dynamics      Optics and Fluid Dynamics    Chaos and Turbulence Studies
and Irreversible Thermodynamics    Risø National Laboratory     Niels Bohr Institute and 
Technical University of Denmark    Building 128                 Department of Chemistry
Building 321                       P.O. Box 49                  University of Copenhagen 
DK-2800 Lyngby                     DK-4000 Roskilde             DK-2100 Copenhagen Ø
Denmark                            Denmark                      Denmark


Jan S. Hesthaven
Division of Applied Mathematics
Brown University, Box F
Providence, RI 02912, USA

Thursday June 28, 2001, 15:00 h
at OFD Meeting Room, Building 130
Risų National Laboratory, 4000 Roskilde

Abstract: The increasing emphasis on the modeling of unsteady problems over very long times and in computationally very large domains requires that accurate and efficient high-order methods be developed for such problems. The geometric complexity, however, of problems of scientific and industrial interest makes such developments a very significant challenge. The use of high-order/spectral methods seems natural due to their superior numerical properties for wave propagation and their highly efficient discretization of large computational domains. However, such methods have traditionally been restricted to domains that can be smoothly mapped to a unit cube, thus allowing for the construction of a well behaved multi-dimensional approximation using censor products. Unfortunately, as is well known, automated grid generation and adaptive meshing is greatly complicated by using only hexahedrals. Guided by these observations we discuss recentdevelopments of accurate high-order/spectral methods on nodal triangles and tetrahedra, on which the approximate solutions are represented using truly multi-variate Lagrange interpolation polynomials. For the approximation of initial boundary value problems we shall discuss how to construct globally stable methods on almost general nodal distributions. The central problem is how to impose the boundary conditions in astable way -- an issue we resolve by imposing the boundary conditions only weakly through a penalty term. As a first example of applications we consider the solution of Maxwell's equations, and we illustrate the performance of the scheme by a number of two- and three-dimensional scattering and penetration problems. For more general conservation laws we shall briefly discuss conservation propertiesand the use of filtering to control the stability of the scheme for general nonlinear problems. We shall briefly illustrate the efficacy of the framework for solving of conservation laws, we show results for he compressible Euler and Navier-Stokes equations for high speed as well as multi-fluid problems.

This work in done in collaboration with Tim Warburton and David Gottlieb at Brown University.