线性化 Painlevé II 和 IV 解的轮廓计算与基于单值性的误差控制
Contour Computation of Linearized Painlevé II and IV Solutions with Monodromy-Based Error Control
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中文总结 AI 辅助
本文提出一种基于单值性误差控制的轮廓积分方法，用于数值求解线性化 Painlevé II 和 IV 方程，并与 Dormand--Prince 方法对比，验证了其在不同背景和区域下的精度与漂移特性。
中文摘要 AI 辅助
我们研究了线性化第二和第四 Painlevé 方程解的轮廓积分表示的数值评估。该构造结合了非线性背景解、规范 Lax 对列的延拓、归一化匹配以及在不同扇区中具有衰减分支的环路上的求积。使用参考基本矩阵、微分方程残差以及独立于谱问题计算的单值性数据的变化来评估精度。一个精确恒等式描述了初始基误差的传播。一种逐阶段后验细化准则利用单值性变化增量来分配数值精度。将轮廓算法与 Dormand--Prince 方法在规则背景上、在快速解增长区域中以及在经历硬稳定性丧失后近似规则振荡的情况下进行了比较。单值性变化中的累积偏移和随后的漂移被分别测量。实验表明漂移如何依赖于谱条件、求积精度以及计算初始部分和后续部分之间的误差分配。另外五个初始数据和参数案例展示了比较性能对非线性背景的依赖性。
英文摘要
We study the numerical evaluation of contour integral representations for solutions of the linearized second and fourth Painlevé equations. The construction combines a nonlinear background solution, continuation of canonical Lax-pair columns, normalization matching, and quadrature over cycles with decaying branches in distinct sectors. Accuracy is assessed using a reference fundamental matrix, differential-equation residuals, and variations of monodromy data computed independently from the spectral problem. An exact identity describes the propagation of initial-basis errors. A stagewise a posteriori refinement criterion uses increments in monodromy variations to allocate numerical accuracy. The contour algorithm is compared with the Dormand--Prince method on a regular background, in a region of rapid solution growth, and through hard loss of stability followed by approximately regular oscillations. Accumulated offsets in monodromy variations and subsequent drift are measured separately. The experiments show how drift depends on spectral conditioning, quadrature accuracy, and error allocation between initial and subsequent parts of the computation. Five additional initial-data and parameter cases demonstrate the dependence of comparative performance on the nonlinear background.
发表机构
- Innopolis University（喀山国立研究信息技术机械学院）
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