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    "title": "Lax and Richtmyer: when a finer numerical mesh approaches the right solution",
    "summary": "For a properly posed linear initial-value problem and a consistent bounded linear difference approximation in the stated Banach-space setting, the1956PartI theorem makes stability necessary and sufficient for convergence for every initial state on a fixed finite time horizon.\n\nMaking a simulation’s grid finer does not by itself make the answer more accurate. The update rule must approximate the right equation, and repeated steps must not amplify some components without a mesh-independent bound. Lax and Richtmyer connect these requirements: within their linear framework, that stability condition is exactly what turns consistency into convergence. Their proof uses uniform boundedness in one direction and accumulated local error plus a dense class of genuine solutions in the other.",
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        "description": "Exact target is 1956 Part I, not the entire 27 page survey. The theorem assumes a complete normed linear space, properly posed linear evolution with a dense unique genuine-solution class and bounded solution extension, bounded whole-space linear updates, a specified joint spatial/time mesh path and consistency uniform in time on a dense genuine-solution class. Convergence is for every initial element as the mesh refines with n*dt tending to a finite time in[0, T]. Stability bounds update powers uniformly over the family/horizon; it is not merely a one-step norm<=1 or eigenvalue test. A specially chosen datum can converge in an unstable scheme without satisfying the all-data statement. The original assumes exact arithmetic and fixed finite horizon, not arbitrary long-time/roundoff guarantees. Part II introduction gives restricted Fourier cases and extra hypotheses for spectral sufficiency; its detailed 277–292 proofs/examples are unread and excluded. Part I 267–275 and bibliography 293 prose/source proof read, formula glyphs imperfect; no independent proof or implementation certification. Nonlinear, adaptive/nonhomogeneous-boundary extensions require their own assumptions.",
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          "citation": "Lax and Richtmyer, Survey of the Stability of Linear Finite Difference Equations (1956):PartI equivalence theorem",
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      "statement": "# Lax and Richtmyer: when a finer numerical mesh approaches the right solution\n\nFor a properly posed linear initial-value problem and a consistent bounded linear difference approximation in the stated Banach-space setting, the1956PartI theorem makes stability necessary and sufficient for convergence for every initial state on a fixed finite time horizon.\n\nMaking a simulation’s grid finer does not by itself make the answer more accurate. The update rule must approximate the right equation, and repeated steps must not amplify some components without a mesh-independent bound. Lax and Richtmyer connect these requirements: within their linear framework, that stability condition is exactly what turns consistency into convergence. Their proof uses uniform boundedness in one direction and accumulated local error plus a dense class of genuine solutions in the other.\n\n## Scientific scope\n\nExact target is 1956 Part I, not the entire 27 page survey. The theorem assumes a complete normed linear space, properly posed linear evolution with a dense unique genuine-solution class and bounded solution extension, bounded whole-space linear updates, a specified joint spatial/time mesh path and consistency uniform in time on a dense genuine-solution class. Convergence is for every initial element as the mesh refines with n*dt tending to a finite time in[0, T]. Stability bounds update powers uniformly over the family/horizon; it is not merely a one-step norm<=1 or eigenvalue test. A specially chosen datum can converge in an unstable scheme without satisfying the all-data statement. The original assumes exact arithmetic and fixed finite horizon, not arbitrary long-time/roundoff guarantees. Part II introduction gives restricted Fourier cases and extra hypotheses for spectral sufficiency; its detailed 277–292 proofs/examples are unread and excluded. Part I 267–275 and bibliography 293 prose/source proof read, formula glyphs imperfect; no independent proof or implementation certification. Nonlinear, adaptive/nonhomogeneous-boundary extensions require their own assumptions.\n\n## People and systems\n\nP. D. Lax; R. D. Richtmyer\n\n\n\n## Original sources\n\n- [Lax and Richtmyer, Survey of the Stability of Linear Finite Difference Equations (1956):PartI equivalence theorem](https://www.coaps.fsu.edu/pub/eric/OCP5930/Papers/StabSurveyLinearFDA.pdf)\n- [Wiley:1956Lax–Richtmyer survey metadata and references](https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160090206)\n",
      "summary": "For a properly posed linear initial-value problem and a consistent bounded linear difference approximation in the stated Banach-space setting, the1956PartI theorem makes stability necessary and sufficient for convergence for every initial state on a fixed finite time horizon.\n\nMaking a simulation’s grid finer does not by itself make the answer more accurate. The update rule must approximate the right equation, and repeated steps must not amplify some components without a mesh-independent bound. Lax and Richtmyer connect these requirements: within their linear framework, that stability condition is exactly what turns consistency into convergence. Their proof uses uniform boundedness in one direction and accumulated local error plus a dense class of genuine solutions in the other.",
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