= of

Several important features are illustrated in Figure 11.11. When c = 1.0, the numerical solution is identical to the exact solution, for the linear convection equation. This is not true for nonlinear PDEs. When c = 0.5, the amplitude of the solution is severely damped as the wave propagates, and the peak of the wave is rounded. The general shape of the solution is maintained, but the leading and trailing edges of the wave are quite smeared out. The result at t = 10.0 s for c = 0.1 is completely smeared out. The numerical solution does not even resemble the general shape of the exact solution. These effects are the result of the numerical damping that is present in the Lax method. In effect, the initialdata distribution is being both convected and diffused, and the effect of diffusion increases as the time step is decreased. The solution for c = 0.9 is much closer to the exact solution, except at the peak, which is severely damped. The presence of large amounts of numerical damping at small values of the convection number, c, is a serious problem with the Lax method.