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      "statement": "# Crank and Nicolson: a practical implicit method for evolving heat flow\n\nCrank and Nicolson develop a time-centred implicit finite-difference procedure for heat-conduction-type equations, combining a stable linear error treatment with practical iteration for nonlinear terms and boundary conditions.\n\nTo calculate how heat spreads, a numerical method advances temperatures from one time to the next. An earlier overlapping-step procedure could amplify tiny alternating errors. Crank and Nicolson instead average spatial differences across the old and new time levels, solving coupled equations at each step. For the linear heat problem this controls error amplitudes without a fixed upper time-step stability limit. Large steps can still leave slowly fading oscillations. Nonlinear heat terms require iteration, while singular starting conditions and narrow rapidly changing regions require careful initialization and resolution. Their comparison of numerical and mechanical procedures made these tradeoffs concrete. The method became a reusable numerical tool; later convergence theorems, fast solvers and remedies for stiff oscillations remain separate contributions.\n\n## Scientific scope\n\nThe centered implicit discretization and nonlinear iterative solve are separate parts of the method. The original error analysis concerns the stated linear heat equation and boundary/symmetry conditions, not universal nonlinear convergence. Large steps can yield slowly decaying oscillations and need not preserve positivity. Singular starting gradients require special treatment. The reviewed source is a permissioned later reprint/transcription of the 1947 work. Reported mesh comparisons and computational costs have not been independently replayed by WRS.\n\n## People and systems\n\nJ. Crank; P. Nicolson\n\n\n\n## Original sources\n\n- [J.Crank and P.Nicolson, practical heat-conduction numerical method — complete1996permissioned reprint of1947work, webOCRtranscription](https://doczz.net/doc/7748002/a-practical-method-for-numerical-evaluation-of-solutions-...)\n- [Publisher reprint notice and twoauthorbyline for Crank/Nicolson original1947work](https://link.springer.com/article/10.1007/BF02127704)\n",
      "summary": "Crank and Nicolson develop a time-centred implicit finite-difference procedure for heat-conduction-type equations, combining a stable linear error treatment with practical iteration for nonlinear terms and boundary conditions.\n\nTo calculate how heat spreads, a numerical method advances temperatures from one time to the next. An earlier overlapping-step procedure could amplify tiny alternating errors. Crank and Nicolson instead average spatial differences across the old and new time levels, solving coupled equations at each step. For the linear heat problem this controls error amplitudes without a fixed upper time-step stability limit. Large steps can still leave slowly fading oscillations. Nonlinear heat terms require iteration, while singular starting conditions and narrow rapidly changing regions require careful initialization and resolution. Their comparison of numerical and mechanical procedures made these tradeoffs concrete. The method became a reusable numerical tool; later convergence theorems, fast solvers and remedies for stiff oscillations remain separate contributions.",
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