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solution are plotted at eleven points on each subinterval.
The rstorder solution (p = 0) shown at the upper left of Figure 10.2.5 is characteristically di usive. The secondorder solution (p = 1) shown at the upper right of Figure
10.2.5 has greatly reduced the di usion while not introducing any spurious oscillations.
The minimum modulus limiter (10.2.7) has attened the solution near the shock as seen
with the thirdorder solution (p = 2) shown at the lower left of Figure 10.2.5. There is
a loss of (local) monotonicity near the shocks. (Average solution values are monotone
and this is all that the limiter (10.2.7) was designed to produce.) The adaptive moment
limiter of Biswas et al. 8] reduces the attening and does a better job of preserving local
monotonicity near discontinuities. The solution with p = 2 using this limiter is shown in
the lower portion of Figure 10.2.5.
Example 10.2.3. Adjerid et al. 2] solve the nonlinear wave equation utt ; uxx = u(2u2 ; 1) (10.2.8a) which can be written in the form (10.1.1a) as
(u1)t + (u1)x = u2 (u2)t ; (u2)x = u1(2u2 ; 1)
1 (10.2.8b) with u1 = u. The initial and boundary conditions are such that the exact solution of
(10.2.8a) is the solitary wave
u(x t) = sech(x cosh 1 + t sinh 1 )
(10.2.8c)
2
2
(cf. Figure 10.2.1).
Adjerid et al. 2] solved problems on ; =3 < x < =3, 0 < t < 1 by the discontinuous Galerkin method using polynomials of degrees p = 0 to 4. The solution at t = 1 10.2. Discontinuous Galerkin Methods 29 Figure 10.2.5: Exact (line) and discontinuous Galerkin solutions of Example 10.2.2 for
p = 0 1 2, and h = 1=32. Solutions with the minmod limiter (10.2.7) and an adaptive
moment limiter of Biswas et al. 8] are shown for p = 2.
performed with p = 2 and J = 64 is shown in Figure 10.2.1. The entire solitary wave is
shown however, the computation was performed on the center region ; =3 < x < =3. 30 Hyperbolic Problems Discretization errors in the L1 norm
J
XZ
ke( t)k = x j j =1 x ;1
j jU (x t) ; Uj (x t)jdx are presented for the solution u for various combinations of h and p in Table 10.2.2.
Solutions of this nonlinear wave propagation problem appear to be converging as O(hp+1)
in the L1 norm. Th...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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