Fast Nyström Methods for Parabolic Boundary Integral Equations
Fast Nyström Methods for Parabolic Boundary Integral Equations
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抛物型边界积分方程的快速 Nyström 方法
DOI：
10.1007/978-3-642-25670-7_6
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
J. Tausch
中科院分区：
文献类型：
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作者：
J. Tausch
抛物型边界积分算子的时间依赖性表现为对前一个问题时间演化的积分形式。内核仅在当前时间是奇异的，并且对于时间上更早的贡献变得越来越平滑。热层势可以看作是广义Abel算子，其核是一个参数依赖的表面积分算子。这种特殊的形式意味着离散化方法和快速评估方法必须从熟悉的椭圆情况下显着改变。在简要回顾了该领域的最新发展之后，我们讨论了在时间上离散Abel积分算子的不同选择。这些方法与标准的表面求积规则相结合，得到抛物型积分方程的Nystrom方法。该方法是明确的，我们将展示如何在空间和时间的快速多极子方法的一个版本可以用来有效地评估时间步进计划。
Time dependence in parabolic boundary integral operators appears in form of an integral over the previous time evolution of the problem. The kernels are singular only at the current time and get increasingly smooth for contributions that are further back in time. The thermal layer potentials can be regarded as generalized Abel operators where the kernel is a parameter dependent surface integral operator. This special form implies that discretization methods and fast evaluation methods must be significantly changed from the familiar elliptic case. After a brief review of recent developments in the area we discuss the different options to discretize Abel integral operators in time. These methods are combined with standard surface quadrature rules to obtain a Nystrom method for parabolic integral equations. The method is explicit and we will show how a version of the fast multipole method in space and time can be used to evaluate the time stepping scheme efficiently.