BAIL 2006 Book of Abstracts - Institut für Numerische und ...

A.E.P. VELDMANN: High-order symmetry-preserving discretization on strongly
stretched grids
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High-order symmetry-preserving discretization on strongly stretched grids
Arthur E.P. Veldman
Institute of Mathematics and Computing Science, University of Groningen
P.O. Box 800, 9700 AV, Groningen, The Netherlands
Many physical phenomena feature thin boundary layers, in which the solution varies much more
rapidly than elsewhere in the domain of interest. A ‘natural’ approach is to adapt a computational grid
to the variations of the solution. In this way one obtains grids with a large diversity in mesh size. Also
the size of adjacent mesh cells can be quite different, i.e. the grid shows strong stretching.
‘Traditional’ discretization methods (based on Lagrangian interpolation) focus on minimizing local
truncation error, but experience has shown that these methods prefer low grid stretching rates (e.g. [2]
and the refernces therein). The problems arise because this approach does not take into account the
properties of the discrete system matrices that arise after discretization. An alternative approach is to
develop discretization schemes with the properties of the system matrix in mind - a general name for
this philosophy is mimetic discretization. Here certain properties of the analytic operator are mimiced
in their discrete counterpart.
At RuG we have chosen to retain the symmetry properties of the operator, in our application a
combination of convection and diffusion. In particular, we discretize convection such that the discrete
version remains skew-symmetric. An almost immediate consequence is that the system matrix remains
diffusively stable (hence never can become singular) on any grid. In formula: Let the system under study
be given by
dφ
+ Lφ = 0,
dt
then the evolution of the kinetic energy � φ �H= φ∗Hφ, where H represents the local mesh size, is given
by
d
dt � φ �= −φ∗ ((HL) ∗ + HL)φ.
With skew-symmetric convection, the symmetric part (HL) ∗ + (HL) of the system matrix comes only
from diffusion, and the above assertion follows.
Another consequence is that the convective discretization does not produce unphysical numerical diffusion,
which unavoidably will interfere with the physical diffusion. This strategy has been applied e.g.
in direct numerical simulation of turbulent flow, where the small-scale balance between convection and
diffusion is quite delicate [3].
In the presentation we would like to illustrate the performance of various symmetry-preserving finitevolume
methods: second- and fourth-order central schemes (that we apply in DNS), but also higher-order
upwind schemes. It is less known that on contracting and expanding grids the traditional upwind method
produces negative diffusion [4]; a symmetry-preserving upwind version (a suitable combination of skewsymmetric
odd derivatives and symmetric even derivatives) will repair this.
For instance, the fourth-order symmetry-preserving discretization of a first-order derivative ∂φ/∂x
becomes
∂φ
∂x ≈ −φi+2 + 8φi+1 − 8φi−1 + φi−2
.
−xi+2 + 8xi+1 − 8xi−1 + xi−2
After muliplication with the denominator (as is usual in a finite-volume setting), the convective contribution
to the coefficient matrix is clearly seen to become skew symmetric.
As a main example we consider a one-dimensional convection-diffusion equation on the unit interval at
a diffusion coefficient k = 0.001. Our ‘favorite’ grid is the Shishkin grid, consisting of piecewise-uniform
grid regions with an abrupt (very large) change in mesh size (e.g. [1]). Figure 1 shows a comparison between
the individual solutions of second- and fourth-order traditional methods and symmetry-preserving
methods on an abrupt Shishkin grid and on a (smoothly stretched) exponential grid. Both grids contain
Speaker: VELDMANN, A.E.P. 152 BAIL 2006
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