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I then evaluate for each sink particle the acceleration error δ a ≡ | a fmm − a true | (9).
If further assuming virial equilibrium and a relative acceleration error ε, | δ E tot | / | E tot | ∼ η ε / N. (68).
Type-III control loops are capable of achieving perfect tracking of step, ramp and parabolic reference signals with zero steady state position, velocity and acceleration error.
Wall-clock time versus relative (top) and scaled (bottom) acceleration error for N = 10 7 particles drawn from a Plummer sphere.
A type-II control loop achieves zero steady state position and velocity error, a type-III control loop achieves zero steady state position, velocity and acceleration error and therefore a type-p control loop is expected to track both faster reference signals and eliminate higher order errors at steady state.
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Figure 3 Acceleration errors from the GPU.
Figure 7 Acceleration errors for naive FMM.
That caused the larger velocity and acceleration errors.
Figure 7 plots the resulting distributions of absolute (top), relative (middle), and scaled (bottom) acceleration errors.
The distributions of acceleration errors resulting from using these estimates in equation (16) are shown in red in Figure 8.
Here, the measured non-conservation of energy actually reflects the amplitude of the acceleration errors in an average sense.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com