This activity is focused on high order methods for partial differential
equations, with the aim of calculating highly accurate numerical solutions to
various flow problems. This can be used to obtain reference solutions for the
evaluation of other methods, or to minimize numerical errors in development and
testing of models, e.g. for turbulence.
The group has worked on spectral methods with Galerkin and collocation
formulations, boundary conditions, and domain decomposition, and the current
focus is on spectral element methods. While retaining the high order, spectral
element methods offer more flexibility in terms of geometry and boundary
conditions than pure spectral methods.
In the spectral element method for solving partial differential equations,
the computational domain is divided into a number of elements, and the numerical
solution is represented as a polynomial on each element. The polynomial order is
usually in the range 6-20. The partial differential equation is solved in weak
form, as for a finite element method, and the continuity of the solution over
element interfaces can be imposed strongly or weakly. Solutions of
multi-dimensional problems are represented by tensor products of polynomials on
Our three-dimensional spectral element implementation is based on hexahedral
elements in Cartesian geometry. The formulation is geometrically (all element
vertices match other element vertices) and functionally (all grid points at
element faces are the same from both sides of the face) conforming.
De-aliasing/over-integration is implemented for stabilization of non-linear
effects, as well as polynomial filtering of the solution. The program runs in
parallel by the use of MPI.
The solution method for the Navier-Stokes equation for incompressible,
viscous flow is based on a splitting of the problem using the
Operator-Integration-Factor method, to obtain an explicit problem for the
advection term and an implicit problem for the viscous terms and the pressure
gradient. The implicit problem is split again by a generalized block LU
decomposition to decouple velocity and pressure.
For variable-density problems, the Boussinesq approximation is implemented,
as well as a pseudo-incompressible formulation.
Suggested literature on spectral element methods:
A. T. Patera: "A spectral element method for fluid dynamics: Laminar flow in
a channel expansion", J. Comput. Phys., vol. 54, pp. 468-488, 1984.
Yvon Maday and Anthony T. Patera: "Spectral element methods for the
incompressible Navier-Stokes equations, in "State of the Art Surveys in
Computational Mechanics", editors A. K. Noor and J. T. Oden, pp. 71-143, ASME,
M. O. Deville, P. F. Fischer, E. H. Mund: "High-Order Methods for
Incompressible Fluid Flow", Cambridge University Press, 2002.
George Em Karniadakis and Spencer J. Sherwin: "Spectral/hp Element Methods
for Computational Fluid Dynamics, second edition", Oxford Science Publications,
C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang: "Spectral Methods -
Evolution to Complex Geometries and Applications to Fluid Dynamics",