Slope Limiters ============== Slope limiters are a critical component of a discontinuous Galerkin method. High order numerical methods applied to non-smooth problems (e.g., effectively all interesting hydrodynamic phenomena) are prone to Gibbs phenomena -- artificial oscillations in the solution. It is the responsibility of a scheme to prevent spurious oscillations while not limiting smooth, physical extrema. Athelas damps spurious oscillations in the discontinuous Galerkin solution with *slope limiters*, applied to the modal (Legendre) coefficients of each field after every explicit stage. A limiter is selected per physics block (``fluid`` and ``radiation``) in the input deck; see `Input Deck`_ below. Setting ``enabled = false`` installs a no-op limiter. Limiting is never applied to a first-order (``basis.nnodes = 1``) run. Available limiters ------------------ ``minmod`` -- TVD minmod The classic Cockburn & Shu TVD(B) limiter. It limits the cell *slope* (mode ``k = 1``) with a minmod of the slope against the forward/backward differences of the neighboring cell averages, and **zeros every higher mode** when the slope is touched. Robust, but it clips smooth extrema and degrades the order of the scheme when it activates. ``moment`` -- hierarchical moment limiter Limits from the highest mode downward and stops as soon as a mode is left unchanged, so it leaves resolved smooth data alone while still controlling oscillations at discontinuities. ``weno`` -- WENO Weighted essentially non-oscillatory reconstruction. (Currently unreliable; prefer ``minmod`` or ``moment``.) The fluid limiters support optional characteristic-variable limiting (``characteristic = true``). Radiation characteristic limiting is currently unsupported. All limiters support a troubled-cell indicator (``tci_opt = true``) that restricts limiting to flagged cells. The TVD minmod limiter ---------------------- The minmod limiter is the total-variation-diminishing slope limiter of the Runge-Kutta DG framework of `Cockburn & Shu (1989) `__, with the TVB relaxation of `Shu (1987) `__. Writing the cell average as :math:`\bar{u}_i = u_i^{(0)}` and the slope coefficient as :math:`u_i^{(1)}`, the limited slope on a non-uniform mesh is .. math:: \tilde{u}_i^{(1)} = \widetilde{m}\!\left( u_i^{(1)},\; b\,{h_i \over 2}\,{(\bar{u}_{i+1} - \bar{u}_i) \over d_{i,+}},\; b\,{h_i \over 2}\,{(\bar{u}_i - \bar{u}_{i-1}) \over d_{i,-}} \right), where :math:`h_i = \Delta x_i` is the local cell width, :math:`d_{i,+} = (h_i + h_{i+1})/2`, and :math:`d_{i,-} = (h_i + h_{i-1})/2`. Neighbor differences are first formed as physical slopes between cell centroids and then rescaled by the half width of the target cell. The factor :math:`b` is the compression parameter (``b_tvd``), and the TVB-modified minmod function is .. math:: \widetilde{m}(a_1, a_2, a_3) = \begin{cases} a_1 & \text{if } |a_1| \le M\,h_i^2, \\ \mathrm{minmod}(a_1, a_2, a_3) & \text{otherwise,} \end{cases} with the standard minmod returning :math:`s\,\min_j |a_j|` when all three arguments share the sign :math:`s` and zero otherwise. The threshold :math:`M` (``m_tvb``) is the TVB constant: when it is comparable to the local second derivative of the solution the limiter leaves genuine smooth extrema untouched, recovering full accuracy there; with :math:`M = 0` the scheme is strictly TVD and clips extrema. When the slope is modified the limiter discards every higher mode (:math:`k \ge 2`), so a limited cell falls back to a (limited) linear profile. This local-width form is what makes the limiter appropriate for the moving Lagrangian mesh, where neighboring zones generally do not have equal widths. The hierarchical moment limiter ------------------------------- The moment limiter follows `Krivodonova (2007) `__. Writing the solution on a cell as an expansion in Legendre modes :math:`u_i = \sum_k u_i^{(k)} P_k`, it limits the coefficients from the highest mode down to the slope :math:`k = 1`, comparing mode :math:`k` against the differences of the *next-lower* mode in the neighboring cells: .. math:: \tilde{u}_i^{(k)} = \widetilde{m}\!\left( u_i^{(k)},\; b\,\beta_k\,{h_i \over 2}\, {(u_{i+1}^{(k-1)} - u_i^{(k-1)}) \over d_{i,+}},\; b\,\beta_k\,{h_i \over 2}\, {(u_i^{(k-1)} - u_{i-1}^{(k-1)}) \over d_{i,-}} \right). Here :math:`\beta_k = 1/(2k-1)` for the un-normalized Legendre coefficients. On a uniform mesh the half-width scaling gives the usual inter-mode factor :math:`1/[2(2k-1)]`. At :math:`k = 1`, :math:`\beta_k = 1`, so this reduces exactly to the TVD minmod slope step above. The cascade stops the moment a coefficient is returned unchanged: if the highest mode is already acceptable the cell is deemed smooth and **all** lower modes are kept. This is the key difference from the TVD minmod limiter, which unconditionally discards every mode above the slope. By limiting only the modes that fail the hierarchical test, the moment limiter preserves high-order accuracy at smooth extrema. The ``b_tvd`` and ``m_tvb`` controls carry the same meaning as in the TVD minmod limiter; the TVB threshold is applied to each modal coefficient with the local width :math:`h_i`. The cell average (:math:`k = 0`) is never modified, so the scheme remains conservative. Input Deck ---------- A limiter is configured per physics block (here ``fluid``): .. code:: lua config.fluid = { limiter = { enabled = true, type = "moment", -- "minmod", "moment", or "weno" b_tvd = 1.0, m_tvb = 0.0, characteristic = false, tci_opt = false, -- tci_val = 1.0, -- required when tci_opt = true }, } The controls are: * ``b_tvd`` -- minmod compression factor on the neighbor-difference arguments. * ``m_tvb`` -- TVB threshold :math:`M`; coefficients with :math:`|u_i^{(k)}| \le M\,h_i^2` are left untouched, which prevents limiting of genuine smooth extrema. A suitable :math:`M` scales with the second derivative of the solution; too large a value disables limiting at real discontinuities, so it should be tuned per problem. * ``characteristic`` -- limit in characteristic variables rather than conserved variables. This is currently supported for fluid limiting only. * ``tci_opt`` / ``tci_val`` -- enable the troubled-cell indicator and set its threshold, restricting limiting to flagged cells.