论文标题
平面中时间相关域的对流扩散方程的积分方程方法
An integral equation method for the advection-diffusion equation on time-dependent domains in the plane
论文作者
论文摘要
边界积分方法对于求解复杂几何形状上的均质线性恒定系数椭圆偏微分方程，因为它们可以在域的边界上以线性或接近线性的计算成本提供准确的解决方案。但是，这些数值方法并非直接适用于时间依赖的方程，这通常是在科学和工程中出现的。我们使用基于积分方程的求解器来解决此问题，用于在两个空间维度中移动和变形几何形状的对流扩散方程。在这种方法中，采用基于半尺寸光谱递延校正的自适应高阶准确时间步变方案。然后，一个时间步长涉及求解一系列非均匀修饰的Helmholtz方程，这是一种称为椭圆行进的方法。我们的解决方案方法利用了几种最近开发的方法，包括特殊用途正交，功能扩展技术和用于修改的Helmholtz内核的光谱Ewald方法。还要特别注意处理时间依赖的几何形状。通过几个数值示例测试了数值方法，以证明鲁棒性，灵活性和准确性
Boundary integral methods are attractive for solving homogeneous linear constant coefficient elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy
