From 449197409cfd70f5350fa15338664354c03201d1 Mon Sep 17 00:00:00 2001 From: Philippe Helluy Date: Thu, 18 Feb 2021 18:53:23 +0100 Subject: [PATCH] up --- mhd/doc/spring_mhd_lbm.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/mhd/doc/spring_mhd_lbm.tex b/mhd/doc/spring_mhd_lbm.tex index 6bc1a5e..9ebab14 100644 --- a/mhd/doc/spring_mhd_lbm.tex +++ b/mhd/doc/spring_mhd_lbm.tex @@ -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 \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. -- GitLab