02-EFT.WP.Core.Equations v1.1 | Chapter 9 — Numerical Realization and Consistency

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Chapter 9 — Numerical Realization and Consistency

I. Aims and Principles

• Establish the minimal closed loop from equations to implementation: discretize → assemble → constrain → solve → evaluate metrics → regress.
• Use the I20-* interfaces as the sole landing path; all formulas and symbols follow Core.Terms and the conventions defined earlier in this volume.
• Define consistency through three classes of metrics: L2, L_inf, T_arr (see Chapters 2 and 8).

II. Grid and Discrete Operators (S90-1)

• Discrete inner products and norms
  inner_h[u,v] def= ( ∑_{i ∈ cells} u_i * v_i * |V_i| )；
  L2_h[u] def= sqrt( inner_h[u,u] )；L_inf_h[u] def= max_i |u_i|。
• Discrete gradient/divergence consistency
  div_h and grad_h must satisfy the discrete Gauss theorem:
  ∑_{i} ( div_h F )_i * |V_i| = ∑_{faces} ( F_f • nu_f ) * A_f。
  Treat this as the acceptance criterion for assemble_operator("div", ...) and assemble_operator("grad", ...).
• S90-1 (discrete conservation constraint)
  For any closed control volume V, if Chapter 6 prescribes zero flux on the boundary, then
  ∑_{i∈V} ( div_h F )_i * |V_i| = 0。

III. Path Discretization and Arrival-Time Consistency (S90-2)

• Line elements and samples
  Given a piecewise approximation of gamma(ell) with ds = { ds_i } and samples n_eff_i:
  T_arr_h def= ( ∑_i ( n_eff_i / c_ref ) * ds_i )。
• Interface consistency
  T_arr_h = propagate_time( n_eff_path={ n_eff_i }, ds={ ds_i }, c_ref ) (binding to I20-4)。
  Must agree with the continuous form: T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) (see Chapter 2).
• S90-2 (segment additivity)
  If gamma = ⋃_k gamma_k, then T_arr_h = ( ∑_k T_arr_h^{(k)} )。

IV. Strong → Weak → Algebraic (S90-3)

• Strong template (example): ∂_t q + div( J_q ) = S_q on Ω。
• Weak template (see Chapter 7):
  weak= inner_Ω[w, ∂_t q] + ( ∫_{Ω} grad[w] • J_q d V ) = inner_Ω[w, S_q] + B[w]。
• Algebraic assembly
  Build matrices/operators M (mass), K (diffusion/advection) via assemble_operator; inject boundary terms with bc_dirichlet, bc_neumann:
  M * dq_dt + K(q) = f + b。
• S90-3 (linearization consistency)
  For one Newton step, linearize K(q) ≈ K(q^n) + J(q^n) * ( q^{n+1} - q^n ). The Jacobian J(•) must match the Gâteaux variation of the weak form (see Chapter 7).

V. Time Marching and Stability (S90-4)

• Explicit advance (example)
  q^{n+1} = q^n + Δt * M^{-1} * ( f^n + b^n - K(q^n) )。
  Convective CFL bound: Δt ≤ CFL * min_i( Δx_i / |u|_{max,i} ), with CFL ∈ (0,1]。
  Diffusive bound: Δt ≤ σ * min_i( Δx_i^2 / ( 2 * d * κ_{max,i} ) ), σ ∈ (0,1], d is spatial dimension.
• Implicit advance (example)
  Solve q^{n+1} from M * ( q^{n+1} - q^n ) / Δt + K( q^{n+1} ) = f^{n+1} + b^{n+1} using solve_nonlinear。
• S90-4 (energy bound)
  For steady diffusion-type K positive definite, implicit Euler satisfies discrete energy decay:
  inner_h[ q^{n+1}, q^{n+1} ] ≤ inner_h[ q^n, q^n ]。

VI. Consistent Injection of Boundary and Initial Data (S90-5)

• Initial condition injection
  Map q^0 = IC(x) to grid nodes/cell centers; if the weak form has a mass matrix, set M * q^0 = inner_Ω[φ_i, IC]。
• Dirichlet boundary
  Use bc_dirichlet(mask, value) for hard replacement or penalty weak enforcement; annotate the chosen method in the equation notes.
• Neumann boundary
  Use bc_neumann(mask, flux) to inject ∫_{∂Ω_N} w * flux d A into the right-hand side b。
• S90-5 (boundary priority)
  Where both value and flux are provided at the same location, prefer Dirichlet as in Chapter 6; Neumann becomes a diagnostic and is not assembled.

VII. Error Metrics and Conservation Diagnostics (S90-6)

• Difference metrics
  E_L2 def= L2_h[ x - y ]；E_Linf def= L_inf_h[ x - y ]。
  Path metric: E_T def= | T_arr(x) - T_arr(y) |, where T_arr(•) is computed by propagate_time。
• Conservation residual
  For a closed system:
  C_err^n def= ( ∑_i q_i^{n+1} * |V_i| - ∑_i q_i^n * |V_i| ) - Δt * ( ∑_i S_{q,i}^n * |V_i| )，expect C_err^n approx 0。
• S90-6 (consistency thresholds)
  Pass if E_L2 ≤ τ_L2 and E_Linf ≤ τ_Linf and E_T ≤ τ_T; thresholds must be stated explicitly in the test plan.

VIII. Grid and Time-Step Convergence (S90-7)

• Order estimate
  With grid sequence h_k = h_0 / 2^k and error E_k = L2_h[ q_{h_k} - q_{h_{k+1}} ], define
  p_est def= log2( E_k / E_{k+1} )。
• S90-7 (order acceptance)
  If theoretical order is p_th, accept when | p_est - p_th | ≤ ε_p, with recommended ε_p ∈ [0.1, 0.3]。

IX. Realizing Statistics and Coarse-Graining in Computation (S90-8)

• Time-average discretization
  avg_t[q; Δt] ≈ ( 1 / N_t ) * ( ∑_{k=0}^{N_t-1} q^{n-k} )，N_t = round( Δt / Δt_num )。
• Volume-average discretization
  avg_V[q; V] ≈ ( 1 / |V| ) * ( ∑_{c ∈ V} q_c * |V_c| )。
• Flux-closure injection
  If using J_sgs^q approx - K_sgs^q * grad[ bar_q ] (see Chapter 8), include K_sgs^q in the diffusion coefficient table coeffs during assembly.

X. Minimal Working Sequence (I20-aligned)*

• Use register_equation(code, eqn, kind, anchors, depends) to register the target equation (see each chapter’s Sxx-*).
• Prepare the grid and path; call discretize_path(gamma, scheme, h) to obtain ds and nodes.
• Build operators M, K, etc., with assemble_operator; inject boundaries with bc_dirichlet and bc_neumann.
• Set IC to obtain q^0; choose a time integrator: solve_linear or solve_nonlinear.
• If arrival time is involved, call propagate_time(n_eff_path, ds, c_ref) to compute T_arr。
• Use compare_solutions(x, y, metrics=["L2","L_inf","T_arr"]) to evaluate differences versus the baseline.
• Record E_L2, E_Linf, E_T, C_err; issue pass/fail per S90-6/7。
• Archive all run metadata (grid, step sizes, thresholds) via export_equations("yaml") and external logs.

XI. Regression and Baselines (S90-9)

• Reference solution
  y_ref comes from analytics, grid extrapolation, or a high-accuracy computation; for path quantities, also store T_arr_ref。
• S90-9 (regression rule)
  pass if ( L2 ≤ τ_L2 and L_inf ≤ τ_Linf ) and | T_arr - T_arr_ref | ≤ τ_T；
  on failure, output the minimal counterexample: grid level, time step, boundary combination, and the corresponding residuals.

XII. Typical Anomalies and Diagnostic Card

• Dimensional check failure: check_dim_equation(eqn) not [target] → roll back to the equation definition and verify coefficients/units (see Core.Terms §6).
• Path not explicit: missing gamma(ell) or d ell → reject (see Chapter 2 and Chapter 8 Lint).
• Conserved quantity drift: nonzero C_err^n accumulates → inspect bc_* injections, div_h/grad_h pairing, and the time integrator.
• Order collapse: p_est far below p_th → examine grid quality, boundary-layer treatment, and subgrid-flux parameters.

XIII. Lint and Forbidden List

• Do not write ∫ n d ell / c or ∫ n_eff / c_ref d ell (missing parentheses or path); the canonical form is ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell )。
• Do not mix n with n_eff or T_fil with T_trans; do not use bare c, T, n。
• Do not omit window/volume parameters: avg_t[•; Δt], avg_V[•; V] must be explicit.

XIV. Registrable Items and Examples

• register_equation("S90-1","discrete inner products and Gauss consistency","definition",anchors=["§II"],depends=["S50-*","S60-*"])
• register_equation("S90-2","discrete arrival time T_arr_h = sum_i (n_eff_i / c_ref) * ds_i","identity",anchors=["§III"],depends=["S20-*","I20-4"])
• register_equation("S90-3","assembly M dq/dt + K(q) = f + b with weak consistency","procedure",anchors=["§IV"],depends=["S70-*"])
• register_equation("S90-4","time stepping stability bounds (CFL/diffusion)","guideline",anchors=["§V"],depends=["S50-*"])
• register_equation("S90-6","error metrics and conservation residuals","metric",anchors=["§VII"],depends=["I20-5"])
• register_equation("S90-7","order-of-accuracy estimation p_est = log2(E_k/E_{k+1})","metric",anchors=["§VIII"],depends=[])
• register_equation("S90-9","regression rule with thresholds on L2, L_inf, T_arr","criterion",anchors=["§XI"],depends=["I20-5","S20-*"])