如何修复不可积的Kahan离散化
How one can repair non-integrable Kahan discretizations
摘要：
Kahan离散适用于具有二次向量场$\dot{x}=f(x)=Q(x)+Bx+c$的$\mathbb R^n$上的任何常微分方程组，并根据公式$(\widetilde{x}-x)/ε=Q(x,\widetilde{x})+B(x+\widetilde{x})/2+c$产生双有理映射$x\mapsto \widetilde{x}$，其中$Q(x,\widetilde{x})$是对应于二次形式$Q(x)$的对称双线性形式。当应用于可积系统时，Kahan离散化比人们预期的先验更频繁地保持可积性，但并非总是如此。我们表明，在某些情况下，原始配方未能保持可积性，可以调整系数的Kahan离散化，以确保其可积性。
英文摘要：
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 $(\widetilde{x}-x)/ε=Q(x,\widetilde{x})+B(x+\widetilde{x})/2+c$, 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.
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