How one can repair non-integrable Kahan discretizations
| Autor: | Matteo Petrera, René Zander, Yuri B. Suris |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Pure mathematics Discretization Integrable system Nonlinear Sciences - Exactly Solvable and Integrable Systems Symmetric bilinear form 010102 general mathematics FOS: Physical sciences General Physics and Astronomy Statistical and Nonlinear Physics Mathematical Physics (math-ph) Quadratic form (statistics) 01 natural sciences 010305 fluids & plasmas Quadratic equation Modeling and Simulation Ordinary differential equation 0103 physical sciences Vector field Exactly Solvable and Integrable Systems (nlin.SI) 0101 mathematics Mathematical Physics Mathematics |
| Popis: | Kahan discretization is applicable to any system of ordinary differential equations on $\mathbb R^n$ with a quadratic vector field, $\dot{x}=f(x)=Q(x)+Bx+c$, and produces a birational map $x\mapsto \widetilde{x}$ according to the formula $(\widetilde{x}-x)/\epsilon=Q(x,\widetilde{x})+B(x+\widetilde{x})/2+c$, where $Q(x,\widetilde{x})$ is the symmetric bilinear form corresponding to the quadratic form $Q(x)$. When applied to integrable systems, Kahan discretization preserves integrability much more frequently than one would expect a priori, however not always. We show that in some cases where the original recipe fails to preserve integrability, one can adjust coefficients of the Kahan discretization to ensure its integrability. Comment: 6 pp |
| Databáze: | OpenAIRE |
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