ner_tags: B-Tag O B-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag I-Tag B-Software I-Software O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O O O O O O O O O O O B-Software I-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software I-Software O O O O O B-Model O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O B-Model O O O O O O B-Model O O O O O O O O O
tokens: </inline-formula> and <inline-formula id="FI40"> <alternatives> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="mml_eq62"> <mml:mover accent="true"> <mml:mrow> <mml:mi mathvariant="sans-serif">B</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="false"> ̄</mml:mo> </mml:mrow> </mml:mover> </mml:math> <tex-math id="tex_eq62"></tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" id="img_eq62" mime-subtype="gif" mimetype="image" xlink:href="aa37995-20-eq62.gif"></inline-graphic> </alternatives> </inline-formula><sup>sCOLA</sup>) Poisson solver (). To make sure that differences between Φ sCOLA ( x , a ) and Φ( x , a ) are as small as possible, we make use of a Poisson solver with Dirichlet boundary conditions, instead of assuming periodic boundary conditions. Such a Poisson solver uses discrete sine transforms (DSTs) instead of FFTs, and requires the boundary values of Φ in six planes (west, east, south, north, bottom, top) surrounding the PM grid (see Appendix C ). These planes have a thickness of N ghost cells (depending on the value of the FDA used to approximate the Laplacian); they are represented by hatched regions in Fig. 2 , right panel. At each scale factor a K when the computation of accelerations is needed, the Dirichlet boundary conditions are extracted from the precomputed Φ BCs ( x , a K ) (step C.3., see Sect. 3.3 ). Ideally, Φ BCs ( x , a K ) should be the exact, non-linear gravitational potential in the full volume at a K , Φ( x , a K ). However, knowing this quantity would require having previously run the monolithic simulation in the full volume, which we seek to avoid. In this paper, we rely instead on the linearly-evolving potential (LEP) approximation ( Brainerd et al. 1993 ; Bagla Padmanabhan 1994 ), namely The idea behind this approximation is that the gravitational potential is dominated by long-wavelength modes, and therefore it ought to obey linear perturbation theory to a better approximation than the density field. In Eq. ( 23 ), we have assumed that the linear growth factor D 1 is normalised to unity at the scale factor corresponding to the initial conditions. The precomputation of Φ BCs in step C.3. is therefore an interpolation from the LPT grid to the PM grid and a simple scaling with D 1 ( a K ). The output of the Poisson solver is the gravitational potential Φ sCOLA ( x , a K ) on the PM grid, in the interior of the sCOLA box (dark green grid points in Fig. 2 , right panel). Consistently with the treatment above, Φ sCOLA ( x , a K ) is padded using the values of Φ BCs ( x , a K ) in 2 N ghost cells around the PM grid, in each direction (light green and yellow regions in Fig. 2 , right panel). Therefore, the only difference between and Δ x resides in using the LEP instead of the true, non-linear gravitational potential at the boundaries of the sCOLA box. Accelerations ( and sCOLA ). Given the gravitational potential Φ sCOLA ( x , a K ), accelerations are computed