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**Figure 1.1.** Dust density snapshots for Problem BA at a grid resolution of 512².
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-512_snapshots.png' caption='**Figure 1.1.** Dust density snapshots for Problem BA at a grid resolution of 512².
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Each row shows the code labeled along the left margin.
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In alphabetical order from top to bottom, the upper group implements an average of one (1) Lagrangian particle per grid cell, and the lower group implements a pressureless dust fluid.
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Each column shows the simulation time in local orbital periods $$T$$ labeled along the top margin.
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The color scale in the bottom right shows the dust density $$\rho_\mathrm{d}$$ relative to the initially uniform gas density $$\rho_\mathrm{g,0}$$ and applies to all snapshots.
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Radial $$x$$ and vertical $$z$$ coordinates are in units of the vertical gas scale height $$H_\mathrm{g}$$.
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The color scale at the bottom right shows the dust density $$\rho_\mathrm{d}$$ relative to the initially uniform gas density $$\rho_\mathrm{g,0}$$ and applies to all snapshots.
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Radial $$x$$ and vertical $$z$$ coordinates are in units of the vertical gas scale height $$H_\mathrm{g}$$.' %}
**Figure 1.2.** Maximum dust densities $$\max(\rho_\mathrm{d})$$ as a function of time for Problem BA at a grid resolution of 512².
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-512_time_series.png' caption='**Figure 1.2.** Maximum dust densities $$\max(\rho_\mathrm{d})$$ as a function of time for Problem BA at a grid resolution of 512².
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The units for density and time are the initially uniform gas density $$\rho_\mathrm{g,0}$$ and the local orbital period $$T$$, respectively.
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The top panel uses a similar cadence for each time series.
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The bottom panel shows the simple moving average $$\mathrm{SMA}$$ with a sampling window of $$10T$$.
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Dotted lines show codes that implement an average of $$n_\mathrm{p} = 1$$ Lagrangian particle per grid cell, solid lines show codes with pressureless dust fluids, and line colors show different codes.
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Dotted lines show codes that implement an average of $$n_\mathrm{p} = 1$$ Lagrangian particle per grid cell, solid lines show codes with pressureless dust fluids, and line colors show different codes.' %}
**Figure 1.3.** Cumulative distribution functions of the dust density for Problem BA at a grid resolution of 512².
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-512_CDF.png' caption='**Figure 1.3.** Cumulative distribution functions of the dust density for Problem BA at a grid resolution of 512².
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In the upper panel, lines show densities time-averaged over the saturation state $$\overline{\rho_\mathrm{d}}$$, line colors different codes, and shaded areas the $$1\sigma$$ time variability.
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In the lower panel, lines and shaded areas show the meta-average $$\overline{\overline{\rho_\mathrm{d}}}$$ and standard deviation, respectively, of $$\overline{\rho_\mathrm{d}}$$ in the upper panel for each dust implementation.
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Dust densities are relative to the initially uniform gas density $$\rho_\mathrm{g,0}$$.
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Dotted lines show codes that implement an average of $$n_\mathrm{p} = 1$$ Lagrangian particle per grid cell and solid lines codes with pressureless dust fluids.
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Dotted lines show codes that implement an average of $$n_\mathrm{p} = 1$$ Lagrangian particle per grid cell and solid lines codes with pressureless dust fluids.' %}
**Figure 1.4.** Similar to Figure [1](#higher-particle-resolution:~:text=Permalink-,Figure%201.,-Dust%20density%20snapshots) except for only codes that implement an average of $$n_\mathrm{p} = 9$$ Lagrangian particles per grid cell.
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-512-np9_snapshots.png' caption='**Figure 1.4.** Similar to Figure [1.1](#higher-particle-resolution:~:text=Permalink-,Figure%201.1.,-Dust%20density%20snapshots) except for only codes that implement an average of $$n_\mathrm{p} = 9$$ Lagrangian particles per grid cell.' %}
**Figure 1.5.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) except comparing only Lagrangian-particle implementations with dashed lines showing an average of $$n_\mathrm{p} = 9$$ particles per grid cell.
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-512-np9_time_series.png' caption='**Figure 1.5.** Similar to Figure [1.2](#higher-particle-resolution:~:text=.-,Figure%201.2.,-Maximum%20dust%20densities) except comparing only Lagrangian-particle implementations with dashed lines showing an average of $$n_\mathrm{p} = 9$$ particles per grid cell.' %}
**Figure 1.6.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative) except comparing only Lagrangian-particle implementations with dashed lines showing an average of $$n_\mathrm{p} = 9$$ particles per grid cell.
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-512-np9_CDF.png' caption='**Figure 1.6.** Similar to Figure [1.3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%201.3.,-Cumulative%20distribution%20functions) except comparing only Lagrangian-particle implementations with dashed lines showing an average of $$n_\mathrm{p} = 9$$ particles per grid cell.' %}
**Figure 1.7.** Similar to Figure [1](#higher-particle-resolution:~:text=Permalink-,Figure%201.,-Dust%20density%20snapshots) except at a grid resolution of 1024².
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-1024_snapshots.png' caption='**Figure 1.7.** Similar to Figure [1.1](#higher-particle-resolution:~:text=Permalink-,Figure%201.1.,-Dust%20density%20snapshots) except at a grid resolution of 1024².' %}
**Figure 1.8.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) except with dashed lines showing grid resolutions of 1024².
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-1024_time_series.png' caption='**Figure 1.8.** Similar to Figure [1.2](#higher-particle-resolution:~:text=.-,Figure%201.2.,-Maximum%20dust%20densities) except with dashed lines showing grid resolutions of 1024².' %}
**Figure 1.9.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative), except grid resolutions at 1024² appear as dashed and darker lines in the upper and lower panels, respectively.
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{% include figure image_path='/assets/images/research/code-comparison/si/BA-1024_CDF.png' caption='**Figure 1.9.** Similar to Figure [1.3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%201.3.,-Cumulative%20distribution%20functions), except grid resolutions at 1024² appear as dashed and darker lines in the upper and lower panels, respectively.' %}
**Figure 2.1.** Similar to Figure [1](#higher-particle-resolution:~:text=Permalink-,Figure%201.,-Dust%20density%20snapshots) except for Problem AB at a grid resolution of 1024².
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{% include figure image_path='/assets/images/research/code-comparison/si/AB-1024_snapshots.png' caption='**Figure 2.1.** Similar to Figure [1.1](#higher-particle-resolution:~:text=Permalink-,Figure%201.1.,-Dust%20density%20snapshots) except for Problem AB at a grid resolution of 1024².' %}
**Figure 2.2.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) except for Problem AB at a grid resolutions of 1024².
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{% include figure image_path='/assets/images/research/code-comparison/si/AB-1024_time_series.png' caption='**Figure 2.2.** Similar to Figure [1.2](#higher-particle-resolution:~:text=.-,Figure%201.2.,-Maximum%20dust%20densities) except for Problem AB at a grid resolutions of 1024².' %}
**Figure 2.3.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative), except for Problem AB at a grid resolutions of 1024².
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{% include figure image_path='/assets/images/research/code-comparison/si/AB-1024_CDF.png' caption='**Figure 2.3.** Similar to Figure [1.3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%201.3.,-Cumulative%20distribution%20functions), except for Problem AB at a grid resolutions of 1024².' %}
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