Convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations
Convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations
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DOI：
10.1002/mma.9795
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发表时间：
2023-11
影响因子：
2.9
通讯作者：
Rui Zhan;Jinwei Fang
中科院分区：
文献类型：
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作者：
Rui Zhan;Jinwei Fang
在本文中，我们将刚性半线性时滞微分方程解的指数积分器的研究推广到非线性情形。除了刚度，还有两个新的问题需要妥善处理：非线性项和时滞项。对于非线性问题，选择不当的线性化可能会导致严重的步长限制。在这项工作中，我们在每一步沿着数值解线性化方程。对于延迟项，采用基于网格点数值而不是级内值的插值法，大大减少了硬阶条件的个数。重点研究了非线性刚性延迟微分方程组的高阶指数Rosenbrock方法的构造和收敛分析。本文的主要结果是，在强连续半群的框架下，证明了即使p$$p$$阶的序条件是弱形式的，显式指数Rosenbrock方法也是p$$p$$阶的刚性收敛的。此外，通过指出不存在小于等于四个阶段的五阶方法，给出了五个阶段的五阶方法的构造。最后，通过数值试验对理论结果进行了验证，验证了高阶方法的优越性。
In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order p$$ p $$ even if the order conditions of order p$$ p $$ hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods.