up

mhd/doc/mhd_lbm.lyx

@@ -2736,12 +2736,578 @@ literal "true"
  solver.
  Here we also used this test to evaluate the efficiency of the parallel
  implementation.
+ The test case is built upon a single vortex, which is a stationary solution
+ of the MHD system, to which a constant drift velocity is added.
+ In the moving frame centered on 
+\begin_inset Formula $\mathbf{r}_{O}(t)=t\mathbf{u}_{\text{\text{drift}}}$
+\end_inset
+
+, with 
+\begin_inset Formula $\mathbf{u}_{\text{\text{drift}}}\in\mathbb{R}^{2}$
+\end_inset
+
+, the analytical solution reads in polar coordinates
+\begin_inset Formula 
+\[
+\begin{align*}\rho(r,\theta) & =\rho_{0},\\
+\mathbf{u}(r,\theta) & =u_{0}[\mathbf{u}_{\text{\text{drift}}}+h(r)\mathbf{e}_{\theta}],\\
+\mathbf{B}(r,\theta) & =b_{0}h(r)\mathbf{e}_{\theta},\\
+p(r,\theta) & =p_{0}+\frac{b_{0}^{2}}{2}(1-h(r)),
+\end{align*}
+\]
+
+\end_inset
+
+with 
+\begin_inset Formula $b_{0}=\rho_{0}u_{0}^{2}$
+\end_inset
+
+.
+ The results shown below are obtained with 
+\begin_inset Formula $\gamma=5/3$
+\end_inset
+
+ and the parameter set 
+\begin_inset Formula 
+\[
+\rho_{0}=p_{0}=1,\ u_{0}=b_{0}=0.2,\ \mathbf{u}_{\text{drift}}=(1,1)^{T},\ h(r)=\exp[(1-r^{2})/2]
+\]
+
+\end_inset
+
+.
 \end_layout
 
 \begin_layout Standard
+The computational domain is the square 
+\begin_inset Formula $\Omega=]-L/2,L/2[\times]L/2,L2[$
+\end_inset
+
+, with 
+\begin_inset Formula $L=10$
+\end_inset
+
+.
+ We compute the solution at time 
+\begin_inset Formula $t=5$
+\end_inset
+
+.
+ The grid contains 
+\begin_inset Formula $N$
+\end_inset
+
+ points in 
+\begin_inset Formula $x_{1}$
+\end_inset
+
+ and 
+\begin_inset Formula $x_{2}$
+\end_inset
+
+ directions.
+ For this smooth test case, we can take a relaxation parameter 
+\begin_inset Formula $\omega=2$
+\end_inset
+
+.
+ We compute the error 
+\begin_inset Formula $e_{N}$
+\end_inset
+
+ in the normalized 
+\begin_inset Formula $L^{1}$
+\end_inset
+
+ norm at the final time between the exact solution and the numerical solution
+ on the first component of the momentum (the other components of the solution
+ would give similar results):
+\begin_inset Formula 
+\[
+e_{N}=\frac{\int_{\Omega}\left|(\rho u_{1})_{\text{num}}-(\rho u_{1})_{\text{exact}}\right|}{L^{2}}.
+\]
+
+\end_inset
+
+Asymptotically, the order of the scheme is evaluated by
+\begin_inset Formula 
+\[
+\beta\simeq\frac{\ln(e_{N}/e_{2N})}{\ln2}.
+\]
+
+\end_inset
+
+The obtained numerical results are summed up in Table 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "tab:Convergence-and-performance"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, where we give the convergence study and a performance evaluation of the
+ implementation.
+ OpenCL permits to run the same code on a multicore CPU or a GPU.
+ It is also possible to adjust the number of used cores in the CPU version.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float table
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+\noindent
+\align center
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="7">
+<features tabularvalignment="middle">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<column alignment="center" valignment="top">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+N
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+128
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+256
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+512
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1024
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2048
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+4096
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+1 CPU
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+16 CPU
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+GPU
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $L^{1}$
+\end_inset
+
+ error 
+\begin_inset Formula $e_{N}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+conv.
+ rate 
+\begin_inset Formula $\beta$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Convergence and performance study
+\begin_inset CommandInset label
+LatexCommand label
+name "tab:Convergence-and-performance"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
 
-\color blue
-TODO: convergence and accuracy test, perfs table.
 \end_layout
 
 \begin_layout Subsection

mhd/vortex_kernels.cl

@@ -177,7 +177,7 @@ void exact_sol(real *x, real t, real *w) {
 
 real gauss(real r){
 
-  return r < 6 ? exp(-r*r/2) : 0;
+  return exp(-r*r/2);
   
 }