Meshfree WENO

A weighted essentially non-oscillatory scheme for scalar equations. In each direction it blends a central MLS gradient with an upwind-biased one; the weights favour whichever stencil is smoother.

From finite volumes to particles

Grid-based WENO combines several candidate stencils with non-linear weights that vanish for stencils crossing a discontinuity. On a particle cloud the candidates are subsets of the neighbourhood:

• Central stencil $C$: all neighbours.
• Directional stencil $S_d$: for each direction $d$, only the neighbours on the upwind side, i.e. $x_{ij,d} < 0$ if $v_d > 0$ and $x_{ij,d} \ge 0$ otherwise.

Each stencil gets an MLS fit of the state differences, and the derivatives of that fit serve as smoothness indicators.

Formulation

For a stencil with fitted derivatives $D^\beta u$ (basis index $\beta$), the smoothness indicator is

\[IS = \sum_{\beta \le D} \big(D^\beta u\big)^2 \Delta x^2 + \sum_{\beta > D} \big(D^\beta u\big)^2 \Delta x^4 ,\]

so first derivatives are weighted by $\Delta x^2$ and all higher-order coefficients by $\Delta x^4$, with $\Delta x = \min_d \Delta x_d$. The weights are

\[\beta = \frac{1/2}{(IS + \epsilon)^2}, \qquad w_S = \frac{\beta_S}{\beta_S + \beta_C}, \quad w_C = \frac{\beta_C}{\beta_S + \beta_C},\]

with $\epsilon = 10^{-12}$, applied per state component. The divergence is

\[\nabla\cdot F_i \approx \sum_d v_d \left( w_S\, (\partial_d u)_{S_d} + w_C\, (\partial_d u)_C \right), \qquad v_d = {\tt velocity}(eq, U_i, d).\]

This is the advective form $v\cdot\nabla u$ with the speed frozen at $U_i$. It is exact for linear advection; for non-linear scalar laws it is a quasi-linear approximation and not conservative.

Construction

WENO(T, D, M, order; div_order = 0)

• order ≥ 2 is required, because the smoothness indicators need second-order coefficients.
• M == 1 is asserted. The upwind side of a stencil needs a single wave speed per direction; systems would need a characteristic decomposition first.
• div_order > 0 caps the fit degree at min(div_order, current order).

Behaviour

• Stateless: no pre-gather pass; the masked fits run inside the flux call and use ib.mask.
• Starved stencils. The central degree is lowered locally until the neighbourhood suffices. A directional stencil that is too small for degree 1 is skipped, and that direction uses the central gradient alone.
• MOOD. The scheme reads particle_orders[i]; _order_reach = 0.
• Moving grids. With an ALEFrame, velocity returns $v_d - w_d$, so both the upwind side and the transport speed are relative to the moving particle.

Reference

WENO{D, M, T, MAX_ORDER, DIV_ORDER, INTERPS} <: DivergenceInterpolator
WENO(::Type{T}, dimension::Int, M::Int, order::Int; div_order::Int=0)
(weno::WENO)(eq, i, f_i, nb_slice, pg, ib, strategy)

WENO(::Type{T}, dimension::Int, M::Int, order::Int; div_order::Int=0)

(weno::WENO)(eq, i, f_i, nb_slice, pg, ib, strategy) -> State{M, T}