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A题/数值分析检验/Prompt/3.md

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+TASK: Produce a deterministic function-level design for simulation with RK4 + nested CPL.
+
+INPUT DATA:
+- MODEL_SPEC from Prompt 1
+- TTE_SPEC from Prompt 2
+- Scenario definition: provides u(t) = [L(t),C(t),N(t),Ψ(t),T_a(t)] for any t
+- Initial state x0 = [z0, v_p0, T_b0, S0, w0]
+- Fixed constants: dt, t_max
+
+METHODOLOGY:
+Define these pure functions (no side effects):
+1) params_to_constitutive(x, params):
+   returns V_oc, R0, Q_eff at current state (with guards z_eff, floors)
+2) power_mapping(u, x, params):
+   returns P_tot
+3) current_cpl(V_oc, v_p, R0, P_tot):
+   returns Δ and I using the specified quadratic root
+4) rhs(t, x, u, params):
+   computes dx/dt using I(t) found by CPL closure
+
+RK4 step (must be spelled out exactly):
+Given (t_n, x_n):
+- Compute u_n = scenario.u(t_n)
+- Stage 1 uses rhs(t_n, x_n, u_n)
+- Stage 2 uses rhs(t_n+dt/2, x_n + dt*k1/2, u(t_n+dt/2))
+- Stage 3 uses rhs(t_n+dt/2, x_n + dt*k2/2, u(t_n+dt/2))
+- Stage 4 uses rhs(t_n+dt, x_n + dt*k3, u(t_n+dt))
+- x_{n+1} = x_n + dt*(k1 + 2k2 + 2k3 + k4)/6
+After updating, clamp states to bounds (z,S,w) as per MODEL_SPEC.
+
+Event evaluation:
+At each grid point, store V_term, z, Δ.
+After each step, check crossings using TTE_SPEC.
+
+DELIVERABLES:
+A) A complete “SIM_API_v1” specification listing:
+   - Function signatures
+   - Inputs/outputs (including units)
+   - Exactly what arrays are stored each step
+   - Termination output bundle
+B) A single canonical output schema:
+   "trajectory" table columns exactly:
+   t, z, v_p, T_b, S, w, V_oc, R0, Q_eff, P_tot, Δ, I, V_term
+   plus metadata: dt, t_max, termination_reason, t_star, TTE_seconds
+
+VALIDATION:
+- Must state the convergence requirement:
+   step-halving: compare dt vs dt/2 with:
+   max|z_dt - z_dt2| < 1e-4 and relative TTE error < 1% (exactly these thresholds)
+- Must include feasibility guard: if Δ becomes negative at any rhs evaluation, trigger event DELTA_ZERO.
+
+OUTPUT FORMAT:
+Return YAML only with keys: SIM_API_v1, OutputSchema, ValidationPlan.
+No prose.