A multilevel method of nonlinear Galerkin type for the Navier-Stokes equations
Hindawi Publishing Corporation
Mathematical Problems in Engineering
The basic idea of this new method resides in the fact that the major part of the relative information to the solution to calculate is contained in the small modes of a development of Fourier series; the raised modes of which the coefficients associated being small, being negligible to every instant, however, the effect of these modes on a long interval of time is not negligible. The nonlinear Galerkin method proposes economical treatment of these modes that permits, in spite of a simplified calculation, taking into account their interaction correctly with the other modes. After the introduction of this method, we elaborate an efficient strategy for its implementation.
1. Introduction
The numerical integration of the Navier-Stokes equations on large intervals of time yields
new problems and new challenges with which we will be faced in the coming years.
Indeed, the considerable increase in the computing power during the last years makes
it thinkable to solve these equations and similar ones in dynamically nontrivial
situations.
In relation with the recent developments in the theory of dynamical systems and its
application to the theoretical survey of the turbulent phenomena (attractors, inertial
manifolds), new algorithms have been introduced by Foias et al. in [
6
], as well as Marion and
Temam in [12].
These methods of multiresolution, also named nonlinear Galerkin methods,
essentially apply to the approximation of nonlinear dissipative systems, as the equations of
Navier-Stokes. Based on a decomposition of the unknowns, as the velocity field, into
small and large eddies, Foias, Manley, and Temam defined new objects: the approximate
inertial manifolds [
6
]. These manifolds define an adiabatic law, modeling the
interaction of the different structures of the flow, the small structures are in fact expressed as a
nonlinear function of large scales. Moreover, these Manifolds enjoy the property that they
attract all the orbits exponentially fast in time and that they contain the attractor in a thin
neighborhood. They provide a good way to approach the solutions of the Navier-Stokes
equations.
These approximate inertial manifolds are subsets of the phase space and consist of an
approximation form of the small scale equations.
The nonlinear Galerkin method, proposed by Marion and Temam [12], consists of
looking for a solution lying on these specific subsets of the phase space.
The first computational tests of this new method were conducted by Jauberteau [10],
Jauberteau et al. [11] in the bidimensional case, where the exact solutions of the equations
were known, they seem appropriate for long time integration of Navier-Stokes equations.
Numerical simulation of turbulent flows being performed at a small fraction of the
computational effort is usually required by traditional methods, see [
4
].
Our aim in this article is to study the implementation of the nonlinear Galerkin
method in the context of pseudospectral discretization for the three-dimensional
NavierStokes equations. Other aspects of multilevel methods of Galerkin type appear in [
5
] by
Dubois et al.
After describing the method, we report on numerical computations based on this
approach. They show an improvement in stability and precision and a significant gain in
computing time.
The calculations of Examples 1 and 2 have been, respectively, carried out on the Cray-2
and Titan.
2. The nonlinear Galerkin method
In this part, we consider the incompressible flows of which the velocity field u = (u1, u2,
u3) in dimension 3 verifies the Navier-Stokes equations:
?u 1
?t ? ? u + (w ? u) + 2 ?|u|2 + ?p = f ,
? ? u = 0,
u(x, t = 0) = u0(x),
where ? is the kinematic viscosity, w(x, t) = ? ? u(x, t) the vorticity, p the pressure, and
f the external force.
Here, | ? | stands for the Euclidean norm in R3.
Moreover, we impose u and p to be periodic in space. Hence, they can be expanded in
Fourier series, namely,
Assuming that u and p lie in the proper Hilbert spaces and applying Pdiv to the
NavierStokes equations (2.1) can be put then under the following abstract form:
?u
?t ? ? u + B(u, u) = g,
where g = Pdiv f and B(u, u) is a bilinear form defined by
B(u, u) = Pdiv(w ? u)
=
k?Z3
k
(w ? u)k ? |k|2 k ? (w ? u)k eik?x.
The numerical procedures are directly applied to this last form of the Navier-Stokes
equations. This formulation is very useful in practice and allows to reduce the memory size of
the codes.
Based on the limit conditions, it is natural to approach (2.6) by a pseudospectral
Galerkin method [
2
], based on a development of u in Fourier series.
We introduce the following decomposition:
where
PN1 and QN1 are operators of projection onto the space of Fourier. yN1 represents the
large scales (structures) of the flow, zN1 the small scales.
After projection of (2.6) on the spaces PN1 and QN1 , the variables yN1 and zN1 are then
solution of the coupled system (...truncated)