High order splitting methods for analytic semigroups exist
High order splitting methods for analytic semigroups exist
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DOI：
10.1007/s10543-009-0236-x
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发表时间：
2009-09-01
影响因子：
1.5
通讯作者：
Ostermann, Alexander
中科院分区：
文献类型：
--
作者：
Hansen, Eskil;Ostermann, Alexander
本文研究了由解析半群演化的抽象发展方程的时间积分的高阶指数分裂方法的构造和分析。本文基于复系数导出了一类新的三至十四阶分裂方法。当应用于具有无界向量场的Banach空间上的方程时，给出了该方法的最优收敛性分析。这些结果解决了半群上是否存在收敛速度大于2的分裂格式的问题。作为一个具体的应用，我们考虑抛物型方程及其维数分裂。数值实验进一步说明了我们的理论误差界的尖锐性。
In this paper, we are concerned with the construction and analysis of high order exponential splitting methods for the time integration of abstract evolution equations which are evolved by analytic semigroups. We derive a new class of splitting methods of orders three to fourteen based on complex coefficients. An optimal convergence analysis is presented for the methods when applied to equations on Banach spaces with unbounded vector fields. These results resolve the open question whether there exist splitting schemes with convergence rates greater then two in the context of semigroups. As a concrete application we consider parabolic equations and their dimension splittings. The sharpness of our theoretical error bounds is further illustrated by numerical experiments.