Use simple average when computing cell-centered B

[felker]

Description

The current second-order accurate conversion from face-averaged to cell-centered B field

 void Field::CalculateCellCenteredField(const FaceField &bf, AthenaArray<Real> &bc,
             Coordinates *pco, int is, int ie, int js, int je, int ks, int ke)

references the volume centered and cell face and volume center positions to compute the weighted average coefficients, for example:

lw = (x2f_jp - x2v_j)/dx2_j;
rw = (x2v_j  - x2f_j)/dx2_j;

which should reduce to 0.5 for uniform Cartesian coordinates. However, due to floating point round-off error, x2f(j+1) - x2v(j) != dx2f(j)/2.0 e.g., so the coefficients lw, rw wont add up to 1.0 exactly. Note for uniform meshes, dx2f = dx2v exactly, but the positions are inexact.

This PR simply uses the existing uniform_limiter[3] switches in the Reconstruction class to determine if the simple averaging formula can be used to compute the cell-centered magnetic field from the face-averaged magnetic fields in each direction.

Testing and validation

mhd-rotor-2D-nx1-32-ic-Bcc1-master.pdf plots the current nx1=nx2=20 with NGHOST=2 initial condition Bcc1 for the 2D MHD rotor problem. The ghost zones are also shown. The initial condition is supposed to have uniform B1 corresponding to bx0=1.410474 requested in the athinput.rotor file, but banding of deviations appear in x1 ranging from min/max:

1.410473999999998
1.4104740000000013

Since the fixes in #98, the deviations are symmetric about the midplane.

mhd-rotor-2D-nx1-32-ic-Bcc1-fix.pdf shows the uniform Bcc1 after the adding the switch in this PR, where all zones real and ghost have

1.4104740000000000

equivalent to 0552802248421E0 in binary.

To-do

• Fix remaining reflective symmetry violations for double precision MHD in Athena++