What is recorded here?
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. Making 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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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.
Read the original work
- Lax and Richtmyer, Survey of the Stability of Linear Finite Difference Equations (1956):PartI equivalence theorem
- Wiley:1956Lax–Richtmyer survey metadata and references
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People and systems
P. D. Lax; R. D. Richtmyer
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English catalogue narrative
Lax and Richtmyer: when a finer numerical mesh approaches the right solution
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.
Making 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.
Scientific scope
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.
People and systems
P. D. Lax; R. D. Richtmyer