Non-uniform meshes

Note

This extension is described and tested in `Benítez-Llambay et al. (2023) `.

Notes about the non-uniform meshes feature:

  • Meshes are created via a mesh-density functions.

  • The feature is enabled by setting SPACING N in the .par file.

  • So far, mesh-density functions are implemented on X and Y only.

  • Non-uniform meshes in azimuth cannot use FARGO advection. Use -DRAM in the .opt instead, which enables

the Rapid Advection Algorithm on Arbitrary Meshes (RAM). RAM is a time-explicit method to solve the advection equation in problems with large bulk velocity on arbitrary computational grids.

Warning

Current implementation of non-uniform meshes is not tested with MHD enabled. It may work with -DSTANDARD, but be careful if you use it.

Warning

Current implementation of the MHD solver does not include RAM.

To test out non-uniform meshes and RAM, the fargo_nu setup is included. This is a slightly modified version of the fargo setup with non-uniform spacing and -DRAM enabled in the .opt file.

First run:

$: make SETUP=fargo_nu
$: ./fargo3d setups/fargo_nu/fargo_nu.par

As an example, we show a snapshot after 50 orbits.

_images/fargo_nu.png

The white box corresponds to the transition between resolution levels. It spans a width YMB-YMA and XMB-XMA in the radial and azimuthal direction, respectively.

Mesh-density functions implemented

Mesh density functions are implemented in src/mesh.c and src/defines.h, and are created in src/LowTasks.c.

Azimuthal and radial spacing are generated via the mesh-density functions described by Eq. (B7) y (B14) in Benítez-Llambay et al. (2023).

Along the azimuthal direction, current implementation assumes the domain to be periodic. Three parameters (in the .par file) control the grid:

  • XMB: it controls where the lowest resolution level changes to the next level.

  • XMA: it controls where the next level is effectively reached.

  • XMC: it controls the resolution ratio between levels.

Notes about the utilization of these parameters:

  • In azimuth the mesh has to be centered at the origin.

  • -XMB has to be greater than XMIN.

  • XMB has to be smaller than XMAX.

  • XMB, XMA must be set such that XMA<XMB.

  • Low-resolution region spans within [XMIN,-XMB] and [XMB,XMAX].

  • High-resolution region spans within the range [-XMA,XMA].

  • The transition between resolution levels occur over the width XMB-XMA.

Along the radial direction, four parameters control the grid spacing:

  • YMA: analog to XMA

  • YMB: analog to XMB

  • YMC: analog to XMC

  • YMY0: it controls the centering of the mesh

In addition, the radial spacing is constructed such that radial spacing is uniform in logarithmic space for |y-YMY0|>YMB.

Implementing a new mesh-density function

To implement a new mesh-density function, the following modifications are needed:

  • Modify the functions ux(), uy() in src/mesh.c. These functions are

the cumulative integrals of the mesh-density functions, and are required to generate the mesh through a bisection algorithm (see src/LowTasks.c, function InitSpace()).

  • Modify the macro UX in src/defines.h so it matches the definition of ux in src/mesh.c.

  • Consider if you need to pass parameters to the cumulative function. In the current version,

we pass parameters as global parameters to ux to save computing time. These globals are set by the function compute_ux_constants(), and are later used by the macro UX (see define.h).

Warning

RAM calls UX both on CPU and GPU to calculate indices efficiently on non-uniform spaces (see src/ram_comput_ustar.c), so consider optimizing any new function implemented.