Discretization rules
Stencils, coefficients, and convergence of the seeded ESD rule files.
The seeded rule files. Each page renders:
- The stencil as a coefficient diagram (which neighbors the rule reads and what coefficient each one carries).
- A convergence plot showing the rule’s empirical order of accuracy on a manufactured solution. Rules whose Layer-B fixtures depend on in-flight ESS harness extensions show a pending placeholder until those fixtures land.
- centered_2nd_uniform
finite_difference
cartesian
scheme
O(dx²)
Two-point centered second-order finite difference for ∂u/∂x on a uniform Cartesian axis.
- centered_2nd_uniform_latlon
finite_difference
latlon
scheme
O(h²)
Centered second-order finite difference on a regular lat-lon grid, with the standard cos λ metric on the longitudinal stencil.
- centered_2nd_uniform_vertical
finite_difference
vertical
scheme
O(h²)
Two-point centered second-order finite difference for ∂u/∂k on a uniformly-spaced vertical axis.
- covariant_laplacian_cubed_sphere
finite_difference
cubed_sphere
scheme
O(h²)
9-point covariant Laplacian on the gnomonic cubed sphere — orthogonal + cross-metric corrections from the inverse metric tensor.
- divergence_arakawa_c
finite_volume
arakawa
scheme
O(h²)
Two-point centered finite-volume divergence on a C-grid — F_x at face_x, F_y at face_y, divergence emitted at cell_center.
- flux_limiter_minmod
limiter
cartesian
limiter
O(dx²) in smooth monotone regions; O(dx) at extrema
Minmod TVD flux limiter (Roe 1986) — the most diffusive of the symmetric second-order TVD limiters.
- flux_limiter_superbee
limiter
cartesian
limiter
O(dx²) in smooth monotone regions; compressive near discontinuities
Superbee TVD flux limiter (Roe 1986) — sits on the upper edge of the second-order TVD region.
- nn_diffusion_mpas
finite_difference
mpas
scheme
O(h²) on quasi-uniform Voronoi
Nearest-neighbor diffusion on an MPAS Voronoi mesh, summed over edges_on_cell with dv_edge / (dc_edge · area_cell) coefficients.
- ppm_reconstruction
finite_volume
cartesian
scheme
O(dx³)
Piecewise-parabolic method (Colella & Woodward 1984) — 4th-order edge values + a parabola per cell, no limiter.
- upwind_1st
finite_difference
cartesian
scheme
O(dx)
First-order upwind (backward) finite difference for ∂u/∂x on a uniform Cartesian axis.
- weno5_advection
finite_volume
cartesian
scheme
O(dx⁵)
Jiang–Shu 5th-order WENO reconstruction — three sub-stencils, smoothness indicators, nonlinear weights.
- weno5_advection_2d
finite_volume
cartesian
scheme
O(h⁵) per axis
Dimension-by-dimension Jiang–Shu WENO5 reconstruction on a 2D structured uniform Cartesian grid.