Numerics

Outline numerics used in the code

*FIXME* is a node-based, collocated finite difference code and the derivatives are to be evaluate on the nodes, based on nodal values. Let be a vector of field values to be differentiated and let be its derivative. The differentiation occurs along coordinate lines using the generic stencil

where are the coefficients of the particular scheme being used. For `SPACEDISC = EXPLICIT`

we take . The coefficients are specified in ModInput.f90. For boundary schemes, biased stencils are used whose coefficients are also specified in ModInput.f90. Many schemes can be found in Lele[1].

**Todo:**- Find specific functions for coefficients

Following Visbal & Gaitonde[3] we compute the metrics in a manner to (mostly) preserves the geometric conservation law. The manner in which the metrics are computed depends on the problem dimension. For 2D we have

while for 3D we have

for the first coodinate,

for the second coodinate, and

for the third coodinate.

In both cases the inverse Jacobian is

with .