论文标题
如何维修不可汇总的Kahan离散
How one can repair non-integrable Kahan discretizations
论文作者
论文摘要
Kahan离散化适用于带有二次矢量字段的$ \ mathbb r^n $上的任何普通微分方程系统，$ \ dot {x} = f（x）= q（x）+bx+c $，并产生一个Birational Map $ x \ mapsto \ mapsto \ mapsto \ didetiLde {x} $， $（\ widetilde {x} -x）/ε= q（x，x，\ widetilde {x}）+b（x+\ \ \ \ widetilde {x}）/2+c $，其中$ q（x，\ widetilde {x}）$是对称的biLineAR形式与quadratic y quadratic quadratic $ $ q（x）$ q（x）。当应用于集成系统时，Kahan离散化可以比先验地预期的要更频繁地保持集成性，但是并非总是如此。我们表明，在某些情况下，如果原始配方无法保留可集成性，则可以调整Kahan离散化的系数以确保其集成性。
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.
