Unformatted text preview: is can be proven correct for smooth solutions of discontinuous Galerkin
methods 2, 11, 12].
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10 8 6 4 2 0 2 4 6 8 10 Figure 10.2.6: Solution of Example 10.2.3 at t = 1 obtained by the discontinuous Galerkin
method with p = 2 and N = 64. J
8
16
32
64
128
256 p=0
2.16e01
1.19e01
6.39e02
3.32e02
1.69e02
8.58e03 p=1
5.12e03
1.19e03
2.88e04
7.06e05
1.74e05
4.34e06 p=2
1.88e04
2.32e05
2.90e06
3.63e07
4.53e08
5.67e09 p=3
7.12e06
4.38e07
2.70e08
1.68e09
1.04e10 p=4
3.67e07
1.12e08
3.55e10
1.10e11
3.49e13 Table 10.2.2: Discretization errors at t = 1 as functions J and p for Example 10.2.3.
Evaluating numerical uxes and using limiting for vector systems is more complicated
than indicated by the previous scalar example. Cockburn and Shu 12] reported problems
when applying limiting componentwise. At the price of additional computation, they
applied limiting to the characteristic elds obtained by diagonalizing the Jacobian fu .
Biswas et al. 8] proceeded in a similar manner. \Fluxvector splitting" may provide a
compromise between the two extremes. As an example, consider the solution and ux
vectors for the onedimensional Euler equations of compressible ow (10.1.3). For this 10.2. Discontinuous Galerkin Methods 31 and related di erential systems, the ux vector is a homogeneous function that may be
expressed as f (u) = Au = fu (u)u:
(10.2.9a)
Since the system is hyperbolic, the Jacobian A may be diagonalized as described in
Section 10.1 to yield f (u) = P;1 Pu
where the diagonal matrix
2
6
=6
6
4 1 contains the eigenvalues of A
3
2
3
u;c
7
2
7=4
5:
u
7
...
5
u+c (10.2.9b) (10.2.9c) m p
The variable c = @p=@ is the speed of sound in the uid. The matrix
decomposed into components = + +; can be
(10.2.10a) where + and ; are, respectively, composed of the nonnegative and nonpositive components of
i j ij i = 1 2 : : : m:
2
Writing the ux vector in similar fashion using (10.2.9)
i = f (u) = P;1( + + ;)Pu = f (u)+ + f (u); : (10.2.10b)
(10.2.10c) Split uxes for the Euler equations were presented by Steger and Warmi...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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