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Unformatted text preview: MATLAB Runge-Kutta 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 piecewise-constant 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
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
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
u0(x t) = 1 1 iif 0=2 x < <=2 :
; f1 x 1
This data is extended periodically to the whole real line. Piecewise-constant 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
convection-dominated convection-di usion equations, some upwinding will be necessary
to eliminate spurious oscillations near discontinuities.
10.2.1 High-Order Discontinuous Galerkin Methods The results of Example 10.2.1 are extreme...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14