AI 中文总结
本文针对变步长对角隐式L(α)-稳定Peer法的传播算子建立严格范数估计，突破Runge-Kutta法的stage阶壁垒，构造出阶数达5的高stage阶Peer法，为刚性半线性问题提供稳定性界并给出对比数值结果。
AI 中文摘要
Peer两步法因兼具Runge-Kutta法与多步法的优良特性（包括高阶精度、强稳定性、避免阶数降低），在刚性初值问题的数值求解中颇具吸引力。本文针对应用于刚性半线性系统的变步长对角隐式L(α)-稳定Peer法的传播算子，建立严格的范数估计。该分析依赖一类Peer法，其传播矩阵拥有与网格无关的左特征向量，可为整个系数族构造合适的矩阵范数。研究表明，这类方法包含 stage 阶q=s的L(α)-稳定Peer法，且s个stage时具有s+1阶超收敛性，突破了不可约s-stage对角隐式Runge-Kutta法的经典stage阶壁垒q≤2，并构造出阶数最高达5的此类方法。实现高stage阶是Peer法的主要优势之一，尤其对刚性微分方程而言，有助于缓解阶数降低问题。所得理论为变网格上的刚性半线性问题提供了一致的稳定性界，为其在最优控制及相关边值问题中的进一步应用奠定基础。本文针对三个经典刚性基准问题给出了对比数值结果，包括三个阶数为3、4、5的新型Peer两步法，以及四个首显式步的stage阶为2、阶数为3、4、5、6的单对角隐式Runge-Kutta法。
英文摘要
Peer two-step methods are attractive for the numerical solution of stiff initial value problems because they combine favorable features of Runge-Kutta and multi-step methods, including high-order accuracy, strong stability properties, and the avoidance of order reduction. This paper develops rigorous norm estimates for the propagation operators of variable-stepsize diagonally-implicit $L(α)$-stable Peer methods applied to stiff semi-linear systems. The analysis relies on a class of Peer methods whose propagation matrices possess a grid-independent left eigenvector, enabling the construction of suitable matrix norms for entire coefficient families. We show that this class contains $L(α)$-stable methods of stage order $q=s$ and super-convergence of order $s+1$ with $s$ stages, exceeding the classical stage-order barrier $q\le 2$ of irreducible $s$-stage diagonally-implicit Runge-Kutta methods, and construct such methods up to order 5. Achieving high stage order is one of the principal advantages of Peer methods, especially for stiff differential equations where it helps mitigate order reduction. The resulting theory provides uniform stability bounds for stiff semi-linear problems on variable grids and establishes a foundation for further applications to optimal control and related boundary value problems. Comparative numerical results are presented for three classical stiff benchmark problems, including three novel Peer two-step methods of order 3, 4, 5, and four singly diagonally implicit Runge-Kutta methods with a first explicit step of stage order 2 and order 3, 4, 5, and 6.
Comments36 pages, 4 figures
