Commit 44919740 authored by Philippe Helluy's avatar Philippe Helluy
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parent 9274cfa1
......@@ -542,11 +542,11 @@ $
In this section, we state some results about the numerical viscosity
of the kinetic relaxation scheme in the one-dimensional case. This
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
$\v F=\v F_{1}$ 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
......@@ -596,7 +596,7 @@ $
\]
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
\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}
......@@ -683,7 +683,7 @@ $
\wvec_{l}=\left(2,\frac{3}{2}\right)^{T}.
\end{align}
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
shows that the matrix relaxation lattice Boltzmann scheme provides
the correct diffusion and therefore results in the right viscous profile.
......
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