September 22, 2016, Revision: 1.4

1 Setup

A wave train with

ln ρ = ux = (A/k) e^{−x^{2}/2σ^{2}} sin(kx)  (1)

is initialized with A = 10^{−4}, σ = 0.5, and k = 10 in a non-periodic domain −π ≤ x ≤ π.

For u_{x} we use an antisymmetric condition, but for the density there is no boundary condition. If we put an antisymmetric condition for ln ρ, we find reflecting waves with almost unchanged amplitude. If we put a symmetric condition, the amplitude decreases significantly with each reflection.

Figure 1: xt diagram of u_{x}(x, t) for ν = 10^{−3} and nx=1024 meshpoints.

2 Questions

What is actually the correct condition for the density in this case?

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Figure 2: Linear momentum evolution using ’a’ for ν = 10^{−3} and nx=1024 meshpoints.

Figure 3: Linear momentum evolution using ’s’ for ν = 10^{−3} and nx=1024 meshpoints.

Figure 4: Linear momentum evolution using ’d1s’ for u_{z} and 1s for ρ for ν = 10^{−3} and nx=1024 meshpoints.