\begin{table}%t4
\caption{$L_1$ norm errors of the density and convergence rate~($r$) under grid
refinement for problem~4 at $t=0.4$. Results for three schemes (Marquina, HLLE,
and Tadmor) and two reconstruction procedures (PPM and~PHM) are shown.}\label{table:convergence}
%\centering
\par
\begin{tabular}{crcccc}
\hline  \hline
 Scheme & N & \multicolumn{2}{c}{$L_1 (10^{-2})$} & \multicolumn{2}{c}{$r$} \\
 \hline
  & & PPM & PHM & PPM & PHM \\
 \hline
  Marquina & 50 & 14.8 & 23.2 & -- & -- \\
           & 100 & 21.0 & 18.5 & --0.50 & 0.30 \\
           & 200 & 16.1 & 15.1 &  0.38 & 0.32 \\
           & 400 & 8.99 & 11.4 & 0.84 & 0.41 \\
           & 800 & 4.22 & 7.22 & 1.09 & 0.66 \\
           & 1600 & 2.22 & 3.87 & 0.93 & 0.90 \\
           & 3200 & 1.04 & 2.06 & 1.09 & 0.91 \\
 \hline
  HLLE & 50 & 14.8 & 23.3 & -- & --\\
       & 100 & 20.9 & 17.0 & --0.50 & 0.29 \\
       & 200 & 16.1 & 15.1 & 0.38 & 0.33 \\
       & 400 & 8.99 & 11.4 & 0.84 & 0.41 \\
       & 800 & 4.22 & 7.22 & 1.09 & 0.66 \\
       & 1600 & 2.22 & 3.87 & 0.93 & 0.90 \\
       & 3200 & 1.04 & 2.06 & 1.09 & 0.91 \\
 \hline 
  Tadmor & 50 & 14.7 & 23.6 &  -- & -- \\
         & 100 & 20.7 & 19.1 & --0.49 & 0.30 \\
         & 200 & 16.0 & 15.2 & 0.37 & 0.33 \\
         & 400 & 8.94 & 11.0 & 0.84 & 0.47 \\
         & 800 & 4.19 & 7.26 & 1.09 & 0.60 \\
         & 1600 & 2.21 & 3.89 & 0.92 & 0.90 \\
         & 3200 & 1.04 & 2.07 & 1.09 & 0.91 \\
 \hline
\end{tabular}
\end{table}