Models A,B, and C: Pulse Profile Evolution

The models we are analyzing are the following. We use pulse profiles and the recommendation of piecewise linear reconstruction with the minimod slope limiter, and the second order TVD Runge-Kutta time stepping.

Model Type	A	B	C
$q_L$	-5	-5	0
$q_C$	-1	5	5
$q_R$	-5	-5	0

The condition given at first, $\lambda=0.5$, was not working for us and after consulting we realized it violates The time step of the numerical scheme must satisfy the Courant-Friedrichs-Lewy (CFL) condition, being small enough to prevent the waves from advancing more than $\Delta x/2$ in $\Delta t$. So we set $\lambda=0.1$ and $\lambda=0.2$ at most. The evolutions can be appreciated in the web site. The time required to evolve has to be increased proportionally to the $\lambda$ modification.

The $l-1 norm$ is also preserved (horizontal line). For A and C models the figure was like (5). for Model B we found a surprise, it was like in figure (6): it is a diagonal line! this does not mean that the conservation of the system is breaking down, but tells us that the L1 method is not suitable for this type of evolution. Physically the area is indeed conserved for B. We have to calculate the area under the curve using some other method.

Figure 5: Monitoring of the L1 Norm conservation.

Figure 6: for Model B we found a surprise, it was It is a diagonal line! this does not mean that the conservation of the system is breaking down, but tells us that the L1 method is not suitable for this type of evolution.

We also investigated the Rankine-Hugoniot condition for the speed of the shock propagation:

$$v_s = \frac{f(q_R-f(q_L)}{q_R-q_L} = (0.5)(q_R+q_L)$$ (35)

This condition is satisfied in all our models. We can appreciate the most extreme behavior in pulse B, where the initial speed of the shock is zero, then the rarefraction that develops causes the amplitude to decrease, and the shock moves to the left. We expect the shock to propagate to the left since $q_R < 0$ is larger than $q_L$ in the long term evolution. satisfied in all three cases, this is most easily seen in case B where, the shock speed is expected to be 0 initially, and we notice when the rarefaction side caused the amplitude to decrease, the shock front begins to propagate to the left, which is expected since $q_R$ which is negative, has a larger magnitude than the $q_L$ value for larger t.