Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

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复杂域上强迫热方程的基于积分方程的数值方法

基于积分方程的数值方法直接适用于均质椭圆PDE，并具有在复杂域上以高精度和高速度求解这些问题的能力。本文将这种方法扩展到具有不均匀源项的热方程。首先，将热量方程式及时离散，然后在每个时间步长中求解一系列所谓的修正的亥姆霍兹方程，其参数取决于时间步长。然后将修改后的Helmholtz方程分为两个部分：用边界积分法求解的齐次部分和特定部分，其中的解决方案是通过评估简单域上非均质源项的体积势来获得的。在这项工作中，我们介绍了两个对成功实现此方法至关重要的组件：