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Numerical Mathematics
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12.04.2011
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The Applied Fluid Mechanics (AFM) group at FFI has a long-term activity on development and implementation of numerical methods.

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 each element.

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, 1989.

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, 2005.

C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang: "Spectral Methods - Evolution to Complex Geometries and Applications to Fluid Dynamics", Springer-Verlag, 2007.

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