22 solutions of this nonlinear wave propagation

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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 2.16e-01 1.19e-01 6.39e-02 3.32e-02 1.69e-02 8.58e-03 p=1 5.12e-03 1.19e-03 2.88e-04 7.06e-05 1.74e-05 4.34e-06 p=2 1.88e-04 2.32e-05 2.90e-06 3.63e-07 4.53e-08 5.67e-09 p=3 7.12e-06 4.38e-07 2.70e-08 1.68e-09 1.04e-10 p=4 3.67e-07 1.12e-08 3.55e-10 1.10e-11 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: (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 non-negative and non-positive 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.

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