22 a 16 element uniform mesh was used and time

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern