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Unformatted text preview: the canonical element as procedure gram(N) N1 := N1=kN1k0 0
for k := 1 toPp do
n
;
t := Nk ; k=11(Nk Ni)0Ni
i
Nk := t=ktk0 0
bf end for return N Figure 10.3.2: GramSchmidt process to construct an orthogonal basis Nk k = 1 2 : : : np
from a basis of monomials Nk , k = 1 2 : : : np . (u v)0 = Z 1 Z 1;
0 0 uvd d kuk 00 = (u u)1=2:
0 (10.3.6a) The result of the GramSchmidt process is a basis Nk , k = 1 2 : : : np that satis es the
orthogonality condition
(Ni Nk ) = ik i k = 1 2 : : : np : (10.3.6b) The actual process can be done using symbolic computation using a computer algebra
system such as MAPLE or MATHEMATICA (cf. Remacle et al. 22] and Problem 2
at the end of this section).
Example 10.3.1. We will illustrate some results using the discontinuous Galerkin
method to solve two and threedimensional compressible ow problems involving the 36 Hyperbolic Problems Euler equations. This complex nonlinear system has the form of (10.3.1a) with
23
2
3
m
n
l
6m7
6 m2 = + p
nm=
lm= 7
67
6
7
7
u=6 n 7
f (u) = f (u) g(u) h(u)] = 6 mn=
n2= + p
ln=
67
6
7
2
4l5
4 ml=
nl=
l = +p 5
e
(e + p)m= (e + p)n= (e + p)b=
23
0
607
67
b(x t u) = 6 0 7 :
(10.3.7a)
67
405
0
Here, is the uid density m, n, and l are the Cartesian components of the momentum
vector per unit volume e is the total energy per unit volume and p is the pressure, which
must satisfy an equation of state of the form p=( ; 1) e ; (m 2 + n2 + l2 )=2 ]: (10.3.7b) This equation of state assumes an ideal uid with gas constant .
Let us consider a classical RayleighTaylor instability which has a heavy ( = 2) uid
above a light ( = 1) uid (Figure 10.3.3). This hydrostatic con guration is unstable and
any slight perturbation will cause the heavier uid to fall and the lighter one to rise. The
uid motion is quite complex and Remacle et al. 22] simulated it using discontinuous
Galerkin methods. They considered twodimensional motion (l = 0, @=@z = 0 in (10.3.7))
with the initial perturbation
1
;y
12
= 1 if 0=2 y < <=2
p = 3=2 ;...
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 Spring '14
 JosephE.Flaherty

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