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140 U. Nackenhorst, M. Ziefle, and A. Suwannachit
Fig. 6 Advection of Gauss profile: Comparison of FD and TDG schemes.
and the analytical solution
α(x, t > 0) = exp
�
�
(x +2− wt)2
− .
2σ
The model parameters are described by a standard deviation of σ =0.05
and a convective velocity of w = 1 m/s. These parameters yield into a very
steep but continuous function with large gradients. Some results are depicted
in Figure 6 to 8 for different schemes and for different Courant numbers. In
Figure 6 concurrent FD schemes are compared with the linear TDG approach
for a subcritical Courant number Cr =0.53. Within the FD family, upstream
and Lax–Wendroff schemes seem to be the most accurate approaches. The
upstream approach is stable, but suffers strongly from diffusion effects, a
much better conservation of the amplitude is observed from Lax–Wendroff
and Leap-Frog methods. However, the latter tend to oscillations. The most
accurate results obtained with FD methods have been for Courant numbers
near to one, from which the requirement of an adaptive time stepping
scheme with respect to the spatial discretization is concluded. In contrast,
the TDG(1) approach shows neither oscillatory nor dispersive behavior.
In contrast to the FD methods the TDG methods are not limited by the
Courant condition (32) and present even for larger time steps (Cr > 1) stable
solutions. The results for TDG schemes of different polynomial order are
compared in Figure 7 for a Courant number Cr =2.66, where the behavior
of improved approximation with increasing polynomial order is evidently
observed.
As an example for the transport of a discontinuous function a square pulse
with infinite gradients is analyzed. With the initial conditions
�
1 : −1 ≤ x ≤ 1,
α(x, t =0)=
0 : else,