Unformatted text preview: MATLAB RungeKutta procedure ode45. The solution
with the upwind ux has greatly dissipated the solution after one period in time. The
maximum error at cell centers
1 0.8 0.6 0.4 U 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.1 0.2 0.3 0.4 0.5
x 0.6 0.7 0.8 0.9 1 Figure 10.2.2: Exact and piecewiseconstant discontinuous solutions of a linear kinematic
wave equation with sinusoidal initial data at t = 1. Solutions with upwind and centered
uxes are shown. The solution using the upwind ux exhibits the most dissipation. je( t)j1 := maxJ ju(xj ; hj =2 t) ; U (xj ; hj =2 t)j
j
1 at t = 1 is shown in Table 10.2.1 on meshes with J = 16, 32, and 64 elements. Since
the errors are decreasing by a factor of two for each mesh doubling, it appears that the 24 Hyperbolic Problems upwind ux solution is converging at a linear rate. Using similar reasoning, the centered
solution appears to converge at a quadratic rate. The errors appear to be smallest at the
downwind (right) end of each element. This superconvergence result has been known for
some time 19] but other more general results were recently discovered 2].
J Upwind Centered
jej1 jej1
16 0.7036 0.1589
32 0.4597 0.0400
64 0.2653 0.0142
Table 10.2.1: Maximum errors for solutions of a linear kinematic wave equation with
sinusoidal initial data at t = 1 using meshes with J = 16, 32, and 64 uniform elements.
Solutions were obtained using upwind and centered uxes.
As a second calculation, let's consider discontinuous initial data
1
u0(x t) = 1 1 iif 0=2 x < <=2 :
; f1 x 1
This data is extended periodically to the whole real line. Piecewiseconstant discontinuous Galerkin solutions with upwind and centered uxes are shown at t = 1 in Figure 10.2.3. The upwind solution has, once again, dissipated the initial square pulse.
This time, however, the centered solution is exhibiting spurious oscillations. As with
convectiondominated convectiondi usion equations, some upwinding will be necessary
to eliminate spurious oscillations near discontinuities.
10.2.1 HighOrder Discontinuous Galerkin Methods The results of Example 10.2.1 are extreme...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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