Solver Numerics

• Default: trapezoidal (good accuracy, A-stable, no numerical damping for oscillatory circuits).
• If numerical ringing after switching transients: switch to backward Euler locally or use LIM (Latency-Insertion Method) damping.
• If stiff system with very different time constants: consider GEAR-2 (BDF-2) for better stability.
• Load references/numerical-integration.md for method equations and companion models.

Reference Guide

Constraints

MUST DO

• Use implicit integration methods (trapezoidal, backward Euler) — explicit methods are unstable for stiff circuits.
• Apply SPICE-style convergence criteria: |Δv_k| < ε_abs + ε_rel·|v_k| for all unknowns.
• Re-factor the matrix whenever the circuit topology changes (switch event or nonlinear re-stamp).
• Track and use the history state (previous inductor currents, capacitor voltages) correctly across time steps.
• Test integration accuracy against known analytical solutions (RC, RL, RLC step response).

MUST NOT DO

• Use forward/explicit Euler for dynamic elements in power circuits — it diverges for practical step sizes.
• Share LU factors across topology changes — the matrix changes, so factors must be recomputed.
• Use absolute convergence criteria only — relative criteria are essential for circuits with large dynamic range.
• Ignore numerical oscillations (ringing) — they indicate method mismatch or step size issue, not circuit behavior.
• Forget to store the integration history state correctly across steps — this is the most common source of wrong transient waveforms.

Output Template

For solver numerics tasks, provide:

1. Integration method in use and why it was chosen (or should be changed).
2. Time step h and its relationship to the fastest dynamics (switching period, LC resonance).
3. Newton-Raphson configuration: max iterations, convergence tolerances, damping strategy.
4. Sparse solver choice and factorization strategy (when to re-factor vs. reuse).
5. Convergence test design: what analytical case to use and what tolerance to expect.
6. Identified numerical problem and proposed fix (oscillation → damping, divergence → smaller step, etc.).

Primary References

• Gear — Numerical Initial Value Problems in Ordinary Differential Equations
• Hairer, Norsett & Wanner — Solving Ordinary Differential Equations I (non-stiff)
• Hairer & Wanner — Solving Ordinary Differential Equations II (stiff, includes BDF)
• Nagel — SPICE2: A Computer Program to Simulate Semiconductor Circuits (UCB/ERL M520)
• Davis — Direct Methods for Sparse Linear Systems (covers UMFPACK, AMD)