Commit 44919740 authored by Philippe Helluy's avatar Philippe Helluy
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...@@ -542,11 +542,11 @@ $ ...@@ -542,11 +542,11 @@ $
In this section, we state some results about the numerical viscosity In this section, we state some results about the numerical viscosity
of the kinetic relaxation scheme in the one-dimensional case. This of the kinetic relaxation scheme in the one-dimensional case. This
one-dimensional analysis will give us simple intuitions for adjusting one-dimensional analysis will give us simple intuitions for adjusting
the relaxation parameter in the 2D case. We consider the one-dimensional the relaxation parameter in the two-dimensional case. We consider the one-dimensional
MHD-DC system (\ref{eq:mhd-1d}). For more simplicity, we will denote MHD-DC system (\ref{eq:mhd-1d}). For more simplicity, we will denote
$\v F=\v F_{1}$ and $x=x_{1}$. The equations then read $\v F=\v F_{1}$ and $x=x_{1}$. The equations then read
\[ \[
\partial_{t}\v w+\partial_{x}\v F(\vw)=0 \partial_{t}\v w+\partial_{x}\v F(\vw)=0.
\] \]
For the analysis, it is possible to replace the scalar relaxation For the analysis, it is possible to replace the scalar relaxation
...@@ -596,7 +596,7 @@ $ ...@@ -596,7 +596,7 @@ $
\] \]
Finally, this analysis provides a way to numerically approximate, Finally, this analysis provides a way to numerically approximate,
up to second order in time the second-order system of conservation up to second order in time, the second-order system of conservation
laws laws
\begin{equation} \begin{equation}
\partial_{t}\vw+\partial_{x}\vF(\vw)-\partial_{x}\left(\v A(\vw)\partial_{x}\vw\right)=0.\label{eq:equiv-eq-2order} \partial_{t}\vw+\partial_{x}\vF(\vw)-\partial_{x}\left(\v A(\vw)\partial_{x}\vw\right)=0.\label{eq:equiv-eq-2order}
...@@ -683,7 +683,7 @@ $ ...@@ -683,7 +683,7 @@ $
\wvec_{l}=\left(2,\frac{3}{2}\right)^{T}. \wvec_{l}=\left(2,\frac{3}{2}\right)^{T}.
\end{align} \end{align}
For the diffusion we use $\epsilon=0.1$. For the Lattice Boltzmann For the diffusion we use $\epsilon=0.1$. For the Lattice Boltzmann
scheme, we set $\lambda=40$. The results on a grid of $128^2$ cells at scheme, we set $\lambda=40$. The results on a grid of $128$ cells at
time $T=0.1$ are shown in Figure \ref{Fg: ViscouseShock}. The test time $T=0.1$ are shown in Figure \ref{Fg: ViscouseShock}. The test
shows that the matrix relaxation lattice Boltzmann scheme provides shows that the matrix relaxation lattice Boltzmann scheme provides
the correct diffusion and therefore results in the right viscous profile. the correct diffusion and therefore results in the right viscous profile.
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