Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions
Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions
复制标题
用于评估二维层热势的混合渐近/数值方法
DOI：
10.1007/s10444-018-9641-5
复制
发表时间：
2018
影响因子：
1.7
通讯作者：
L. Greengard
中科院分区：
文献类型：
--
作者：
Jun Wang;L. Greengard
本文提出了一种精确计算二维单层和双层热势的渐近/数值混合方法。在以前的工作中已经表明，简单的正交方案遭受一种称为“几何诱导刚度”的现象，这意味着在观察到快速收敛率之前，正式的高阶精确方法需要过小的时间步长。这可以通过时间上的解析积分来克服，这需要对一组具有非物理、弱奇异核的空间边界积分算子进行评估。在我们的混合格式中，我们将局部渐近逼近与仅涉及高斯核的几个边界积分算子的求值结合起来，这些算子很容易被新版本的快速高斯变换加速。该方案具有鲁棒性，避免了几何诱导刚度，并且易于在存在运动几何的情况下使用。它扩展到三维是自然和直接的，并且应该允许层热势成为灵活和强大的工具来模拟扩散过程。
We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.