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Unformatted text preview: ly discouraging. It would appear that we have
to contend with either excessive di usion or spurious oscillations. To overcome these
choices, we investigate the use of the higher-order techniques o ered by (10.2.4). With
cij being an m-vector and i ranging from 0 to p, we have p +1 vector and m(p +1) scalar
unknowns on each element.
We will focus on the four major tasks: (i) evaluating the integral on the right side
of (10.2.4a), (ii) performing the time integration (iii) de ning the initial conditions,
and (iv) evaluating the uxes. The integral in (10.2.4a) will typically require numerical
integration and the obvious choice is Gaussian quadrature as described in Chapter 6.
This works ne and there is no need to discuss it further.
Time integration can be performed by either explicit or implicit techniques. The
choice usually depends on the spread of the eigenvalues i, i = 1 2 : : : m, of the Jacobian A(u). If the eigenvalues are close to each other, explicit integration is ne. Stability 10.2. Discontinuous Galerkin Methods 25 2 1.5 1 U 0.5 0 −0.5 −1 −1.5 −2 0 0.1 0.2 0.3 0.4 0.5
x 0.6 0.7 0.8 0.9 1 Figure 10.2.3: Exact and piecewise-constant discontinuous solutions of a linear kinematic
wave equation with discontinuous initial data at t = 1. Solutions with upwind and
centered uxes are shown. The solution using the upwind ux is dissipative. The solution
using the centered ux exhibits spurious oscillations.
is usually not a problem. An implicit scheme might be necessary when the eigenvalues are
widely separated or when integrating (10.2.4) to a steady state. For explicit integration,
Cockburn and Shu 12] recommend a total variation diminishing (TVD) Runge-Kutta
scheme. However, Biswas et al. 8] found that classical Runge-Kutta formulas gave similar results. Second- and third-order and fourth- and fth-order classical Runge-Kutta
software was used for time integration of Example 10.2.1. If forward Euler integration of
(10.2.4a) were used, we would have to solve the explicit system
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- Spring '14