jax homme

Parent post for the jax homme project.

Notes:

  • Prognostic equations use strong derivatives, but Laplacian is calculated weakly in diffusion, even in sweqx.
  • Element access must be order-agnostic outside of boundary exchange routines!
  • Internal grid is constructed

Todo:

  • Create topology and gridgen :((
  • Create structs
  • Implement metric terms + quadrature
  • Implement boundary exchange/DSS
  • Verify boundary exchange/DSS on sphere
  • implement derivatives
  • Implement compute_and_apply_rhs for sweqx.

Gridgen notes:

  • Ordering within a reference element is
      E1
   [v1  v2]
E2 [      ] E3
   [v3  v4]
      E4
  • An unstructured grid is a set of pairs ((pos1,pos2,pos3,pos4),((elemE1,vert idv1,vert idv2),…,(elemE4,vert idv3,vert idv4)))( (\textrm{pos}_1, \textrm{pos}_2, \textrm{pos}_3, \textrm{pos}_4), ((\textrm{elem}_{E_1}, \textrm{vert id}_{v_1}, \textrm{vert id}_{v_2} ), \ldots, (\textrm{elem}_{E_4}, \textrm{vert id}_{v_3}, \textrm{vert id}_{v_4} ))) where posi\textrm{pos}_i are specified as lat-lon points on the sphere.
  • Set of transformations is [−1,1]2↦Cube↦S2[-1, 1]^2 \mapsto \textrm{Cube} \mapsto \mathbb{S}^2
  • Converting this into metric terms for the sphere requires mapping the GLL points in the reference grid to the sphere and taking the product of the consequent Jacobians.
    • Requires DSS of metric terms, as mappings from reference element to cube often results in coincident points having different Jacobian values.
  • Standard quasi-homogeneous cube has topology (assuming dice labeling) of