35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 10 Numerical Methods & Simulation (SimStack-OT)

Chapter 10 Numerical Methods & Simulation (SimStack-OT)

I. Abstract & Scope
This chapter specifies the numerical methods and execution framework of the oriented-tension simulation stack (SimStack-OT): orientation discretization on S^2, time stepping for Q_ij dynamics (S80-4), discretization of the coupling free-energy kernel W_cpl and transport/wave modules, numerical closure of the energy ledger (EDX), and forward mapping with instrument response R_inst plus the synthetic-observation pipeline. All symbols use English notation with backticks; SI units apply. No ToA terms appear here.

II. Dependencies & References

• Geometry & orientation: Chapter 3 (S80-1/2 for S^2 quadrature and Q_ij definitions).
• Axioms & minimal equations: Chapter 4 (P80-2/3/9/10, S80-3/4).
• Metrology: Chapter 5 (M80-1…4).
• Couplings & media: Chapter 6 (S80-5/6).
• Energy accounting: Chapter 7 (S80-7/8).
• Implementation & APIs: Chapter 12 (I80-*).

III. Normative Anchors (added in this chapter, S80-/M80-)

• S80-15 (Consistency of S^2 Orientation Discretization): directions and weights {n_hat^α, w_α} satisfy ∑_α w_α = 1 and the second-moment correction ∑_α w_α ( n_i^α n_j^α − δ_ij/3 ) = 0.
• S80-16 (Projection & Tracelessness of Q_ij): at every time step project the numerical Q_ij to symmetry and tracelessness: Q ← (Q+Q^T)/2 − (Tr(Q)/3) I.
• S80-17 (Energy-Consistent Operator Splitting): advance S80-4 with a three-stage split—advection/rotation, relaxation/diffusion, sources—such that the discrete energy W_orient^h and power terms have errors bounded by O(Δt^p + h^q) and pass the EDX audit.
• S80-18 (CFL & Stability Conditions): time step satisfies Δt ≤ min{ C_adv h/‖u_vec‖_∞ , C_diff h^2/(‖D_Q‖_∞ + ε) }, with C_adv,C_diff ∈ (0,1].
• M80-31 (SimCfg & Module Registry): required fields of the simulation config SimCfg and the modular registration workflow.
• M80-32 (S^2 Grid Generation & Error Assessment): build S^2 grids (HEALPix/Lebedev/equal-area) and report quadrature errors.
• M80-33 (Time Integrators): selection and stability reporting for Runge–Kutta, Strang splitting, and semi-implicit Crank–Nicolson.
• M80-34 (Coupling Kernels & Media Modules): discretizations and interfaces for W_cpl, anisotropic diffusion/waves.
• M80-35 (EDX Ledger Kernel): computation of discrete W_orient, Φ_E, and 𝒫_*, with closure-residual assessment.
• M80-36 (Instrument Response & Sampling): forward mapping with R_inst, noise injection, and synthetic data generation.
• M80-37 (Regression & Benchmarks): fixed-seed benchmark suites and pass/fail thresholds.
• M80-38 (Parallelism & Performance): parallel strategies over orientation/space and scalability reporting.

IV. Body Structure

I. Background & Problem Statement

• SimStack-OT aims to reproduce experimentally observable Q_ij, T_fil_ij, anisotropic transport/wave properties, and the energy ledger within a unified numerical framework, cross-checking with metrology (Ch.5) and coupling (Ch.6).
• Design principles: geometric consistency (S^2), energy consistency (EDX), constraint consistency (symmetry/traceless/positive/objective), and reproducible/auditable artifacts.

II. Key Equations & Derivations (S-series)

1. S80-15 (S^2 Orientation Discretization):
   • Quadrature: ∫_{S^2} g(n_hat) dΩ ≈ ∑_α w_α g(n_hat^α), with ∑ w_α = 1.
   • Second-moment correction: ∑_α w_α ( n_i^α n_j^α − δ_ij/3 ) = 0, ensuring isotropy tests pass.
2. S80-16 (Numerical Projection):
   For any approximate Q^*, set Q ← (Q^*+Q^{*T})/2 − (Tr(Q^*)/3) I to keep Q=Q^T and Tr(Q)=0.
3. S80-17 (Split Advancement):
   • Step 1 (advection/rotation): ∂_t Q + u·∇Q − Ω Q − Q Ω = 0.
   • Step 2 (relaxation/diffusion): ∂_t Q = − (Q−Q^eq)/tau_relax + D_Q ∇^2 Q.
   • Step 3 (sources): ∂_t Q = S_ij(Q, fields).
   • Apply S80-16 after each substep; global error O(Δt^2) for Strang or O(Δt) for first-order splitting.
4. S80-18 (Stability Bounds):
   • Advection CFL: Δt ≤ C_adv h / max‖u_vec‖.
   • Diffusion CFL: Δt ≤ C_diff h^2 / (‖D_Q‖ + ε).
   • Use the more restrictive bound for mixed steps.

III. Methods & Flows (M-series)

1. M80-31 SimCfg & Module Registry
   • Required: {sim_id, seed, geometry, S2_grid, time:{t0,dt,T}, physics:{A,K,tau_relax,D_Q,Λ,χ,α,κ}, media:{ε,μ,C}, coupling:{W_cpl}, edx:{K_E}, instrument:{R_inst,noise}, outputs, units}.
   • Module registration: register(module_name, version, params); produce dependency tables and hashes.
2. M80-32 S^2 Grid & Error
   • Generate {n_hat^α,w_α};
   • Evaluate errors with analytic tests (1, n_i n_j − δ_ij/3);
   • Emit S2Grid.json and an error report.
3. M80-33 Time Integrators
   • Options: RK2/RK3, Strang splitting, CN semi-implicit (diffusion step);
   • Output stability and energy-residual curves (vs S80-7/8).
4. M80-34 Coupling Kernels & Media
   • Discretize W_cpl and D_eff, ε_ij, μ_ij, C_eff;
   • Implement directional projections ê_i Q_ij ê_j and beam/wave-vector grids.
5. M80-35 EDX Ledger Kernel
   • Compute discrete W_orient^h, Φ_E^h, 𝒫_mech^h, 𝒫_cpl^h, 𝒫_diss^h;
   • Closure audit: statistics of r_EDX = ⟨ ∂_t W_orient^h + ∇·Φ_E^h − (𝒫_in^h − 𝒫_diss^h) ⟩.
6. M80-36 Instrument Response & Sampling
   • y_sim = R_inst ⊗ x_sim + n;
   • Support Poisson/Gaussian and correlated-noise injections.
7. M80-37 Regression & Benchmarks
   • S/M/L tasks; fixed-seed artifacts; metric gates (e.g., SpecMAE, EDX-closure, AnisoMap-PSNR);
   • Produce metrics.json and pass/fail flags.
8. M80-38 Parallelism & Performance
   • Orientation-dimension parallelism (direction blocks), spatial domain decomposition (MPI/multiprocessing), vectorization;
   • Report throughput (steps/s), strong/weak scaling, memory peaks.

IV. Cross-References within/beyond this Volume

• Chapter 4: numerical intake of constitutive/dynamic parameters (Λ_{ijkl}, A,K,tau_relax,D_Q).
• Chapter 5: metrology posteriors as initial/boundary conditions and for noise calibration.
• Chapter 6: validation of coupling/anisotropy modules.
• Chapter 7: numerical EDX closure audits and segmented accumulation.
• Chapter 12: I80-* endpoints & released artifacts; benchmark registration & acceptance.
• Companion: numerical & benchmark chapters in Energy Filaments.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • S80-15/16/17/18 satisfied; r_EDX mean near zero within confidence bands.
   • Error orders match theory on analytic/semi-analytic benchmarks (isotropic/weakly anisotropic).
   • Disabling couplings or diffusion worsens evidence and errors.
2. Negative criteria:
   • Long-time drift to Tr(Q)≠0 or Q≠Q^T.
   • CFL violations causing energy blow-up or spurious oscillations.
   • Benchmark metrics below gates without stabilization measures.
3. Contrasts:
   • {RK, splitting, CN} effects on stability and energy closure.
   • {HEALPix, Lebedev, equal-area} quadrature accuracy vs cost.
   • {no coupling, EM-only, mechanical-only, composite} explanatory power for observables.

VI. Deliverables & Figure List

1. Deliverables:
   • SimCfg.json (full configuration & version hashes).
   • S2Grid.json (directions & weights).
   • Products/ (Q(t,r), T(t,r), AnisoMaps, PowerTerms, Ledgers, …).
   • metrics.json (SpecMAE, EDX-closure, AnisoMap-PSNR, throughput & scaling).
   • UnitsAudit.log and RegressionReport.md.
2. Figures/Tables (suggested):
   • Tab. 10-1 Time integrators vs stability/accuracy.
   • Tab. 10-2 S^2 grid errors & second-moment correction results.
   • Fig. 10-1 Energy-closure residual spectra in typical scenarios.
   • Fig. 10-2 Anisotropy maps vs experiments.
   • Tab. 10-3 Benchmark tasks and pass/fail summary.