Graduate School in Nonlinear Science
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
HIGH-ORDER/SPECTRAL UNSTRUCTURED GRID METHODS FOR CONSERVATION LAWS
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
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.