--- name: math-thumb-pde description: "Apply Chapter 23 (Partial Differential Equations: Discretize, Resolve, and Verify) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to compute scheme-specific stability limits and design a refinement check that separates space and time errors." --- # Partial Differential Equations: Discretize, Resolve, and Verify Use this chapter to help the reader make a checked mathematical decision. All 15 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: PDE, coefficients and units; domain and boundary/initial data; spatial and temporal schemes; grid spacing; conservation and error targets. 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 - **Is the PDE transport-dominated?** Follow characteristic direction, inspect cell Peclet numbers, and use limiting near discontinuities. - **Must a quantity balance cell by cell?** Form one shared numerical flux per face and audit the telescoping global balance. - **Is nominal high order missing?** Check solution regularity, geometry, quadrature, and solver tolerance before increasing polynomial degree. - **Is time integration explicit?** Evaluate every applicable advection, diffusion, reaction, and source stability restriction. - **Are waves or thin layers important?** Set the mesh from wavelength and layer thickness, then refine the output that depends on their derivatives or phase. - **Do space and time errors interact?** Isolate each with separate studies before choosing a coupled refinement path. - **Has the code been verified?** Run manufactured solutions and recover predicted order in relevant norms. - **Could the domain or linear solve contaminate the answer?** Move artificial boundaries and tighten multigrid-preconditioned solves independently. 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:** CFL constants depend on dimension and scheme. Numerical stability is not a resolution certificate; verify boundaries, conservation, and order separately. 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 23.2.1, 23.2.2, 23.3.2 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. - **C23-D01: Evolve transport with a chosen CFL number**: rule 23.2.1. - **C23-D02: Execute diffusion against a known sine solution**: rule 23.2.2. - **C23-D03: Measure solution error on successive meshes**: rule 23.3.3. - **C23-D04: See waves drift on a coarse grid**: rule 23.2.3. - **C23-D05: See centered differences wiggle when flow outruns diffusion**: rule 23.1.2. 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.