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Docs/source/theory/amr.rst

@@ -30,7 +30,7 @@ A sketch of the implementation of mesh refinement in WarpX is given in :numref:`
 
    Position history of one charged particle attracted by its image induced by a nearby metallic (dirichlet) boundary. The particle is initialized at rest. Without refinement patch (reference case), the particle is accelerated by its image, is reflected specularly at the wall, then decelerates until it reaches its initial position at rest. If the particle is initialized inside a refinement patch, the particle is initially accelerated toward the wall but is spuriously reflected before it reaches the boundary of the patch whether using the method implemented in WarpX or the MC method. Providing a surrounding transition region 2 or 4 cells wide in which the potential is interpolated from the parent coarse solution reduces significantly the effect of the spurious self-force.
 
-The presence of the self-force is illustrated on a simple test case that was introduced in :cite:t:`amr-Vaylpb2002` and also used in :cite:t:`amr-Colellajcp2010`: a single macroparticle is initialized at rest within a single refinement patch four cells away from the patch refinement boundary. The patch at level :math:`L_1` has :math:`32\times32` cells and is centered relative to the lowest :math:`64\times64` grid at level :math:`L_0` (“main grid”), while the macroparticle is centered in one direction but not in the other. The boundaries of the main grid are perfectly conducting, so that the macroparticle is attracted to the closest wall by its image. Specular reflection is applied when the particle reaches the boundary so that the motion is cyclic. The test was performed with WarpX using either linear or quadratic interpolation when gathering the main grid solution onto the refined patch boundary. It was also performed using another method from P. McCorquodale et al (labeled “MC” in this paper) based on the algorithm given in :cite:t:`amr-Mccorquodalejcp2004`, which employs a more elaborate procedure involving two-ways interpolations between the main grid and the refined patch. A reference case was also run using a single :math:`128\times128` grid with no refined patch, in which it is observed that the particle propagates toward the closest boundary at an accelerated pace, is reflected specularly at the boundary, then slows down until it reaches its initial position at zero velocity. The particle position histories are shown for the various cases in :numref:`fig_ESselfforce`. In all the cases using the refinement patch, the particle was spuriously reflected near the patch boundary and was effectively trapped in the patch. We notice that linear interpolation performs better than quadratic, and that the simple method implemented in WarpX performs better than the other proposed method for this test (see discussion below).
+The presence of the self-force is illustrated on a simple test case that was introduced in :cite:t:`amr-Vaylpb2002` and also used in :cite:t:`amr-Colellajcp2010`: a single macroparticle is initialized at rest within a single refinement patch four cells away from the patch refinement boundary. The patch at level :math:`L_1` has :math:`32\times32` cells and is centered relative to the lowest :math:`64\times64` grid at level :math:`L_0` ("main grid"), while the macroparticle is centered in one direction but not in the other. The boundaries of the main grid are perfectly conducting, so that the macroparticle is attracted to the closest wall by its image. Specular reflection is applied when the particle reaches the boundary so that the motion is cyclic. The test was performed with WarpX using either linear or quadratic interpolation when gathering the main grid solution onto the refined patch boundary. It was also performed using another method from P. McCorquodale et al (labeled "MC" in this paper) based on the algorithm given in :cite:t:`amr-Mccorquodalejcp2004`, which employs a more elaborate procedure involving two-ways interpolations between the main grid and the refined patch. A reference case was also run using a single :math:`128\times128` grid with no refined patch, in which it is observed that the particle propagates toward the closest boundary at an accelerated pace, is reflected specularly at the boundary, then slows down until it reaches its initial position at zero velocity. The particle position histories are shown for the various cases in :numref:`fig_ESselfforce`. In all the cases using the refinement patch, the particle was spuriously reflected near the patch boundary and was effectively trapped in the patch. We notice that linear interpolation performs better than quadratic, and that the simple method implemented in WarpX performs better than the other proposed method for this test (see discussion below).
 
 .. _fig_ESselfforcemap:
 
@@ -43,7 +43,7 @@ The presence of the self-force is illustrated on a simple test case that was int
 The magnitude of the spurious self-force as a function of the macroparticle position was mapped and is shown in :numref:`fig_ESselfforcemap` for the WarpX and MC algorithms using linear or quadratic interpolations between grid levels. It is observed that the magnitude of the spurious self-force decreases rapidly with the distance between the particle and the refined patch boundary, at a rate approaching one order of magnitude per cell for the four cells closest to the boundary and about one order of magnitude per six cells beyond. The method implemented in WarpX offers a weaker spurious force on average and especially at the cells that are the closest to the coarse-fine interface where it is the largest and thus matters most.
 We notice that the magnitude of the spurious self-force depends strongly on the distance to the edge of the patch and to the nodes of the underlying coarse grid, but weakly on the order of deposition and size of the patch.
 
-A method was devised and implemented in WarpX for reducing the magnitude of spurious self-forces near the coarse-fine boundaries as follows. Noting that the coarse grid solution is unaffected by the presence of the patch and is thus free of self-force, extra “transition” cells are added around the “effective” refined area.
+A method was devised and implemented in WarpX for reducing the magnitude of spurious self-forces near the coarse-fine boundaries as follows. Noting that the coarse grid solution is unaffected by the presence of the patch and is thus free of self-force, extra "transition" cells are added around the "effective" refined area.
 Within the effective area, the particles gather the potential in the fine grid. In the extra transition cells surrounding the refinement patch, the force is gathered directly from the coarse grid (an option, which has not yet been implemented, would be to interpolate between the coarse and fine grid field solutions within the transition zone so as to provide continuity of the force experienced by the particles at the interface). The number of cells allocated in the transition zones is controllable by the user in WarpX, giving the opportunity to check whether the spurious self-force is affecting the calculation by repeating it using different thicknesses of the transition zones. The control of the spurious force using the transition zone is illustrated in :numref:`fig_ESselfforce`, where the calculation with WarpX using linear interpolation at the patch interface was repeated using either two or four cells transition regions (measured in refined patch cell units). Using two extra cells allowed for the particle to be free of spurious trapping within the refined area and follow a trajectory that is close to the reference one, and using four extra cells improved further to the point where the resulting trajectory becomes indistinguishable from the reference one.
 We note that an alternative method was devised for reducing the magnitude of self-force near the coarse-fine boundaries for the MC method, by using a special deposition procedure near the interface :cite:p:`amr-Colellajcp2010`.