Revised

_research/05-code-comparison.md

@@ -41,46 +41,45 @@ Further details on the structure and contents of submission data can be found in
 ![](/assets/images/research/code-comparison/si/BA-512_snapshots.png)
 **Figure 1.** Dust density snapshots for Problem BA at a grid resolution of 512².
 Each row shows the code labeled along the left margin.
-In alphabetical order from top to bottom, the upper group implements dust fluids 
-In alphabetical order from top to bottom, the upper group implements Lagrangian particles with $$n_\mathrm{p} = 1$$ ([Problem Set](#problem-set) Section 2.1.2), and the lower group implements dust fluids (Section 2.1.3).
+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.
 Each column shows the simulation time in local orbital periods $$T$$ labeled along the top margin.
-The color scale in the bottom right shows the dust density $$\rho_\mathrm{d}$$ in units of the initially uniform gas density $$\rho_\mathrm{g,0}$$.
+The color scale in the bottom right shows the dust density $$\rho_\mathrm{d}$$ relative to the initially uniform gas density $$\rho_\mathrm{g,0}$$.
 Radial $$x$$ and vertical $$z$$ coordinates are in units of the vertical gas scale height $$H_\mathrm{g}$$.
 
 ![](/assets/images/research/code-comparison/si/BA-512_time_series.png)
 **Figure 2.** Maximum dust densities $$\max(\rho_\mathrm{d})$$ as a function of time for Problem BA at a grid resolution of 512².
-The top panel compares time series with similar cadences, showing an average of 630 samples per code.
+The units for density and time are the initially uniform gas density $$\rho_\mathrm{g,0}$$ and the local orbital period $$T$$, respectively.
+The top panel uses a similar cadence for each time series with an average of 630 samples per code.
 The bottom panel shows the simple moving average $$\mathrm{SMA}$$ with a sampling window of $$10T$$.
-Densities are in units of the initially uniform gas density $$\rho_\mathrm{g,0}$$.
-Dotted lines show codes that implement Lagrangian particles with $$n_\mathrm{p} = 1$$, solid lines show codes with dust fluids, and line colors show different codes.
+Dotted lines show codes that implement an average of one (1) Lagrangian particle per grid cell, solid lines show codes with pressureless dust fluids, and line colors show different codes.
 
 ![](/assets/images/research/code-comparison/si/BA-512_CDF.png)
 **Figure 3.** Time-averaged cumulative distribution functions for the dust density for Problem BA at a grid resolution of 512².
 Lines show the densities time-averaged over the saturation state $$\overline{\rho_\mathrm{d}}$$, and shaded areas show the $$1\sigma$$ time variability.
-Densities are in units of the initially uniform gas density $$\rho_\mathrm{g,0}$$.
-Dotted lines show codes that implement Lagrangian particles with $$n_\mathrm{p} = 1$$, solid lines show codes with dust fluids, and line colors show different codes.
+Dust densities are relative to the initially uniform gas density $$\rho_\mathrm{g,0}$$.
+Dotted lines show codes that implements an average of one (1) Lagrangian particle per grid cell, solid lines show codes with pressureless dust fluids, and line colors show different codes.
 
 
 ##### Higher Particle Resolution
 
 ![](/assets/images/research/code-comparison/si/BA-512-np9_snapshots.png)
-**Figure 4.** Similar to Figure [1](#higher-particle-resolution:~:text=Permalink-,Figure%201.,-Dust%20density%20snapshots) except for only codes with $$n_\mathrm{p} = 9$$ Lagrangian particles.
+**Figure 4.** Similar to Figure [1](#higher-particle-resolution:~:text=Permalink-,Figure%201.,-Dust%20density%20snapshots) except for only codes that implement an average of nine (9) Lagrangian particles per grid cell.
 
 ![](/assets/images/research/code-comparison/si/BA-512-np9_time_series.png)
-**Figure 5.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) except comparing only Lagrangian-particle implementations with dash–dotted lines showing $$n_\mathrm{p} = 9$$.
-The top panel compares time series with similar cadences, showing an average of 2000 samples per code.
+**Figure 5.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) except comparing only Lagrangian-particle implementations and that dash–dotted lines show an average of nine (9) particles per grid cell.
+The top panel uses a similar cadence for each time series with an average of 1000 samples per code.
 
 ![](/assets/images/research/code-comparison/si/BA-512-np9_CDF.png)
-**Figure 6.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative) except comparing only Lagrangian-particle implementations with dash–dotted lines showing $$n_\mathrm{p} = 9$$.
+**Figure 6.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative) except comparing only Lagrangian-particle implementations and that dash–dotted lines show an average of nine (9) particles per grid cell.
 
 #### Higher Grid Resolution
 
 ![](/assets/images/research/code-comparison/si/BA-1024_snapshots.png)
 **Figure 7.** Similar to Figure [1](#higher-particle-resolution:~:text=Permalink-,Figure%201.,-Dust%20density%20snapshots) except at a grid resolution of 1024².
 
 ![](/assets/images/research/code-comparison/si/BA-1024_time_series.png)
-**Figure 8.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) with dash–dotted lines showing grid resolutions of 1024².
-The top panel compares time series with similar cadences, showing an average of 2000 samples per code.
+**Figure 8.** Similar to Figure [2](#higher-particle-resolution:~:text=.-,Figure%202.,-Maximum%20dust%20densities) but with dash–dotted lines showing grid resolutions of 1024².
+The top panel uses a similar cadence for each time series with an average of 2000 samples per code.
 
 ![](/assets/images/research/code-comparison/si/BA-1024_CDF.png)
-**Figure 9.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative) with dash–dotted lines showing grid resolutions of 1024².
+**Figure 9.** Similar to Figure [3](#higher-particle-resolution:~:text=show%20different%20codes.-,Figure%203.,-Time%2Daveraged%20cumulative) but with dash–dotted lines showing grid resolutions of 1024².