Geometric Numerical Integration¶
Hairer, E., Wanner, G., & Lubich, C. (2006). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer.
Cited by¶
6 citations across 6 artifacts.
Each citation links to the sentence it supports in the citing article.
Domain-specific¶
- Butcher Group
- Different texts distribute symmetry and tree-factorial terms between the basis and the coefficient map, so raw coefficients must never be compared across conventions without translation.
This sourceStandard reference for order conditions, trees, B-series, composition, symplectic methods, and backward error analysis.
- Different texts distribute symmetry and tree-factorial terms between the basis and the coefficient map, so raw coefficients must never be compared across conventions without translation.
- Hamiltonian Mechanics
- (an exact lift, not analogy). Numerical integration — symplectic integrators (Verlet, leapfrog) that exploit the preserved symplectic structure for long-time-stable orbits where non-symplectic schemes drift in energy
This sourceA complete theory of symplectic and symmetric numerical methods for Hamiltonian systems, covering the Stormer-Verlet scheme, and showing why preservation of geometric structure yields more accurate long-time integration than general-purpose methods.
Supported in partVerified against the work's full text
“the preservation of geometric properties of the flow not only produces an improved qualitative behaviour, but also allows for a more accurate long-time integration than with general-purpose methods”
- (an exact lift, not analogy). Numerical integration — symplectic integrators (Verlet, leapfrog) that exploit the preserved symplectic structure for long-time-stable orbits where non-symplectic schemes drift in energy
- Symplectic Structure
- … symplectic form on the 6-dimensional phase space), geometric quantisation (the cohomology class of ω determines the prequantum line bundle), and numerical integration (symplectic integrators preserve a nearby ω exactly, preventing the long-term energy drift that afflicts generic integrators in solar-system simulations
This sourceHairer, Lubich and Wanner's Geometric Numerical Integration, whose backward-error analysis shows geometric integrators preserving structure on the level of a modified equation and giving more accurate long-time integration than general-purpose methods.
Supported in partVerified against the source
- … symplectic form on the 6-dimensional phase space), geometric quantisation (the cohomology class of ω determines the prequantum line bundle), and numerical integration (symplectic integrators preserve a nearby ω exactly, preventing the long-term energy drift that afflicts generic integrators in solar-system simulations
- Verlet Integration
Mechanisms¶
- Structure-Preserving Numerical Integration
- What distinguishes it from a merely accurate integrator is that it optimizes for qualitative long-run fidelity over per-step accuracy — it will happily be a little wrong about where a particle is now in exchange for never being wrong about the structure the dynamics live in.
This sourcePrioritizes preservation of geometric structure and qualitative long-time behavior even when pointwise numerical states are approximate.
- What distinguishes it from a merely accurate integrator is that it optimizes for qualitative long-run fidelity over per-step accuracy — it will happily be a little wrong about where a particle is now in exchange for never being wrong about the structure the dynamics live in.
- Symplectic Form Preservation Check
- What sets it apart is its geometric, whole-map stance: it does not manipulate the coordinate functions' brackets but interrogates the derivative matrix, and it treats certifying preservation across the domain as the deliverable — closer to gathering validation evidence than to checking a single identity.
This sourcePresents symplecticity tests through derivative-matrix conditions and treats structure preservation across the numerical map.
- What sets it apart is its geometric, whole-map stance: it does not manipulate the coordinate functions' brackets but interrogates the derivative matrix, and it treats certifying preservation across the domain as the deliverable — closer to gathering validation evidence than to checking a single identity.
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 2 of 6 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
Was it audited? Yes. A second, independent pass read the citation against the article text and recorded a verdict.
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