--- name: math-thumb-ode description: "Apply Chapter 22 (Ordinary Differential Equations: Choosing and Checking Time Integrators) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to separate time-integrator stability from accuracy and verify the quantities the application needs." --- # Ordinary Differential Equations: Choosing and Checking Time Integrators Use this chapter to help the reader make a checked mathematical decision. All 12 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: ODE and initial state; state/time units; shortest important timescale; stiffness clues; component tolerances; events and invariants. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Are accepted explicit steps far smaller than output accuracy seems to require?** Inspect Jacobian scales and solver statistics; test a stiff method at the same tolerances. - **Is the system strongly dissipative and large?** Try BDF with sparse Jacobians. If it is Hamiltonian and long-time geometry matters, choose a symplectic method instead. - **Does the solution oscillate?** Set a starting maximum step from the shortest important period, then verify phase and amplitude under refinement. - **Are state scales heterogeneous or near zero?** Choose componentwise absolute tolerances and a dimensionless relative tolerance before judging cost. - **Does the model switch, impact, or cross a threshold?** Define a smooth event function and localize a bracketed root; do not shrink the display grid as a substitute. - **Is the method explicit?** Check its full stability region against relevant modes, not only its formal order. - **Can the exact solution be avoided?** Use three-level refinement, invariant drift, and balance checks as independent evidence. - **Do implicit solves dominate?** Expose sparsity and escalate to appropriate sparse or matrix-free linear solvers. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Formal order and local error estimates do not replace stability, event, phase, or invariant checks. Use the stability region of the actual integrator. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 22.1.1, 22.2.3, 22.3.1 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C22-D01: Solve a decay problem and watch the stability limit**: rule 22.3.1. - **C22-D02: Verify RK4 by step halving**: rule 22.3.2. - **C22-D03: Check an oscillator through its energy**: rule 22.1.3. - **C22-D04: Watch stiffness force tiny explicit steps**: rule 22.1.1. - **C22-D05: Count steps per period before trusting an oscillation**: rule 22.1.5. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.