Unformatted text preview: is can be proven correct for smooth solutions of discontinuous Galerkin
methods 2, 11, 12].
-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
3.49e-13 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 component-wise. 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. \Flux-vector splitting" may provide a
compromise between the two extremes. As an example, consider the solution and ux
vectors for the one-dimensional 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:
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
4 1 contains the eigenvalues of A
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 non-negative and non-positive components of
i j ij i = 1 2 : : : m:
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