# 22 a 16 element uniform mesh was used and time

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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 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. 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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