All Figures

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5870_f01.ps}\end{figure}$ Figure 1: Relative variation of mass ( top panel) and angular momentum ( middle panel) of the system {gas + Jupiter + outflow} with time. The absolute variation of angular momentum of the system is compared to that of the planet on the bottom panel (in our units, in which the initial angular momentum of the planet is 10^-3).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,angle=270]{5870_f02.ps}\end{figure}$ Figure 2: Relative angular momentum variation as a function of time. The variation of angular momentum of the system $\{$ gas + planet + star + outflow $\}$ , $\delta H$ , is compared to its initial value H₀ (bottom curve, plain line), to the angular momentum of the planet $H_{\rm p}$ (middle curve, long-dashed line), and to the variation of the latter $\delta H_{\rm p}$ (top curve, short-dashed line).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm]{5870_f03.eps}\end{figure}$ Figure 3: Sketch of the coupling via ghost rings at the inner interface between the 1D grid and the 2D grid.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,angle=270]{5870_f04.ps} \end{figure}$ Figure 4: Viscous spreading of a ring. The theoretical curves (obtained by numerical solution of Eq. (14) of Lynden-Bell & Pringle (1974) with the boundary condition $\Sigma (0.02)=0$ ) are thin, while the numerical ones obtained with our code are bold. The vertical dashed lines show the position of the interfaces, where the solution provided by our code remains perfectly smooth.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f05.ps} \end{figure}$ Figure 5: Gas surface density profile after 16 000 time units $\sim$ 2500 orbits of the Jupiter mass planet initially placed at $r_{\rm p}=1$ . The solid line corresponds to a 2D (bold) and a 1D (thin) grid coupled, while the dashed line stands for a 2D grid alone. The dot-dashed line shows the initial profile.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f06.ps} \end{figure}$ Figure 6: Relative variation of mass and angular momentum in the case of a 2D grid with open boundaries and a 2D grid coupled with a 1D grid.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f07.ps} \end{figure}$ Figure 7: Inner disk profiles at the end of simulations with different locations of the inner interface between the 1D and the 2D grid. The bold part of the profiles corresponds to the 2D grid.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f08.ps} \end{figure}$ Figure 8: Inner disk profiles at the end of simulations similar to those of Fig. 7, but without the implementation of the wave damping algorithm (6). A strong sensitivity on $R_{\rm interface}$ appears.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f09.ps} \end{figure}$ Figure 9: Outer disk profile close to the outer interface ( $R_{\rm interface}=2.9167$ , marked by a vertical line) at 3 different times. The vertical line at r=2.3167 marks the outer interface used in a second simulation, whose results are indistinguishable from those plotted in this figure.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.8cm,clip]{5870_f10.ps} \end{figure}$ Figure 10: Type II migration of the Jupiter mass planet in the disk. The semi-major axis is plotted as a function of time for all the cases already discussed: various positions of the inner interface between the two grids, and no 1D grid.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f11.ps}\end{figure}$ Figure 11: Left panel: migration of a Jupiter mass planet initially put on a circular orbit at $r_{\rm p}=1.4$ . Right panel: radial velocity in the gas disk when the planet is introduced.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5870_f01.ps}\end{figure}$	Figure 1: Relative variation of mass ( top panel) and angular momentum ( middle panel) of the system {gas + Jupiter + outflow} with time. The absolute variation of angular momentum of the system is compared to that of the planet on the bottom panel (in our units, in which the initial angular momentum of the planet is 10^-3).
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$\begin{figure} \par\includegraphics[width=8.8cm]{5870_f03.eps}\end{figure}$	Figure 3: Sketch of the coupling via ghost rings at the inner interface between the 1D grid and the 2D grid.
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$\begin{figure} \par\includegraphics[width=8cm,angle=270]{5870_f04.ps} \end{figure}$	Figure 4: Viscous spreading of a ring. The theoretical curves (obtained by numerical solution of Eq. (14) of Lynden-Bell & Pringle (1974) with the boundary condition $\Sigma (0.02)=0$ ) are thin, while the numerical ones obtained with our code are bold. The vertical dashed lines show the position of the interfaces, where the solution provided by our code remains perfectly smooth.
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$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f05.ps} \end{figure}$	Figure 5: Gas surface density profile after 16 000 time units $\sim$ 2500 orbits of the Jupiter mass planet initially placed at $r_{\rm p}=1$ . The solid line corresponds to a 2D (bold) and a 1D (thin) grid coupled, while the dashed line stands for a 2D grid alone. The dot-dashed line shows the initial profile.
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$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f06.ps} \end{figure}$	Figure 6: Relative variation of mass and angular momentum in the case of a 2D grid with open boundaries and a 2D grid coupled with a 1D grid.
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$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f07.ps} \end{figure}$	Figure 7: Inner disk profiles at the end of simulations with different locations of the inner interface between the 1D and the 2D grid. The bold part of the profiles corresponds to the 2D grid.
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$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f08.ps} \end{figure}$	Figure 8: Inner disk profiles at the end of simulations similar to those of Fig. 7, but without the implementation of the wave damping algorithm (6). A strong sensitivity on $R_{\rm interface}$ appears.
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$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f09.ps} \end{figure}$	Figure 9: Outer disk profile close to the outer interface ( $R_{\rm interface}=2.9167$ , marked by a vertical line) at 3 different times. The vertical line at r=2.3167 marks the outer interface used in a second simulation, whose results are indistinguishable from those plotted in this figure.
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$\begin{figure} \par\includegraphics[angle=-90,width=8.8cm,clip]{5870_f10.ps} \end{figure}$	Figure 10: Type II migration of the Jupiter mass planet in the disk. The semi-major axis is plotted as a function of time for all the cases already discussed: various positions of the inner interface between the two grids, and no 1D grid.
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$\begin{figure} \par\includegraphics[width=8.5cm,angle=270]{5870_f11.ps}\end{figure}$	Figure 11: Left panel: migration of a Jupiter mass planet initially put on a circular orbit at $r_{\rm p}=1.4$ . Right panel: radial velocity in the gas disk when the planet is introduced.
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