Error Inhibiting Block One-step Schemes for Ordinary Differential Equations
Error Inhibiting Block One-step Schemes for Ordinary Differential Equations
复制标题
常微分方程的误差抑制块一步法
DOI：
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发表时间：
2017
影响因子：
2.5
通讯作者：
S. Gottlieb
中科院分区：
文献类型：
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作者：
A. Ditkowski;S. Gottlieb
常用的单步法和线性多步法均存在与局部截断误差相同量级的全局误差（定义见[1,6,8,13,15]）。事实上，这对所有的一般线性方法都是成立的。在实践中，这意味着该方法的阶数通常仅由阶数条件定义，而阶数条件是通过研究局部截断误差得出的。在这项工作中，我们研究了局部截断误差和全局误差之间的相互作用，并开发了一种方法，该方法定义了显式误差抑制块一步方法的构造（或者写成显式一般线性方法[2]）。这些误差抑制方案的构建是为了控制局部截断误差随时间的累积，从而导致全局误差比局部截断误差高一个数量级。在这项工作中，我们描述了如何仔细选择系数矩阵，以抑制局部截断误差的增长。然后，我们利用这一理论认识构建了几种具有比局部截断误差更高阶的全局误差的方法，并在测试用例上证明了它们提高了精度的顺序。这些方法表明，误差抑制的概念是可以实现的。未来的工作将进一步开发新的误差抑制方法，并分析这些方法的计算效率和线性稳定性。
The commonly used one step methods and linear multi-step methods all have a global error that is of the same order as the local truncation error (as defined in [1, 6, 8, 13, 15]). In fact, this is true of the entire class of general linear methods. In practice, this means that the order of the method is typically defined solely by order conditions which are derived by studying the local truncation error. In this work we investigate the interplay between the local truncation error and the global error, and develop a methodology which defines the construction of explicit error inhibiting block one-step methods (alternatively written as explicit general linear methods [2]). These error inhibiting schemes are constructed so that the accumulation of the local truncation error over time is controlled, which results in a global error that is one order higher than the local truncation error. In this work, we delineate how to carefully choose the coefficient matrices so that the growth of the local truncation error is inhibited. We then use this theoretical understanding to construct several methods that have higher order global error than local truncation error, and demonstrate their enhanced order of accuracy on test cases. These methods demonstrate that the error inhibiting concept is realizable. Future work will further develop new error inhibiting methods and will analyze the computational efficiency and linear stability properties of these methods.