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- W53150207 abstract "An arbitrary consistent one-step approximation of an ordinary differential equation is studied. If the scheme is assumed to be accurate of O(△t r ) then it may be shown to be O(△t r + m ) accurate as an approximation to a modified equation for any integer m ≥ 1. A technique is introduced for proving that the modified equations inherit qualitative properties from the numerical method. For Hamiltonian problems the modified equation is shown to be Hamiltonian if the numerical method is symplectic, and for general problems the modified equation is shown to possess integrals shared between the numerical method and the underlying system. The technique for proving these results does not use the well-known theory for power series expansions of particular methods such as Runge-Kutta schemes, but instead uses a simple contradiction argument based on the approximation properties of the numerical method. Although the results presented are known, the method and generality of the proofs are new and may be of independent interest." @default.
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- W53150207 date "1997-01-01" @default.
- W53150207 modified "2023-09-23" @default.
- W53150207 title "On the Qualitative Properties of Modified Equations" @default.
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- W53150207 doi "https://doi.org/10.1007/978-3-642-60539-0_12" @default.
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