An efficient high-order meshless method for advection-diffusion equations on time-varying irregular domains
An efficient high-order meshless method for advection-diffusion equations on time-varying irregular domains
复制标题
时变不规则域平流扩散方程的高效高阶无网格方法
DOI：
10.1016/j.jcp.2021.110633
复制
发表时间：
2021
影响因子：
4.1
通讯作者：
Fogelson, Aaron L.
中科院分区：
文献类型：
--
作者：
Shankar, Varun;Wright, Grady B.;Fogelson, Aaron L.
提出了一种高阶径向基函数有限差分（RBF-FD）方法求解时变区域上的对流扩散方程。我们的框架是基于最近开发的重叠RBF-FD方法的推广，该方法利用一种新的自动程序计算RBF-FD在模板中心周围的可变大小区域的stenetry权重。该过程消除了重叠参数δ，从而使得能够在移动域上进行RBF-FD微分矩阵的无调谐组装。此外，我们的框架利用了一个简单而有效的程序更新微分矩阵的移动域平铺节点集随时间变化的基数。最后，对流扩散在时变域处理通过快速节点集修改，一个新的高阶半拉格朗日方法，利用新的调谐自由重叠RBF-FD方法，和高阶时间积分方法的组合。最后得到的框架没有调优参数，时间复杂度为O（N log N）。我们证明了高阶收敛的对流扩散方程的时变二维和三维域的小和大的Peclet数。我们还提出了验证我们的复杂性估计的时间。最后，我们利用我们的方法来解决一个耦合的3D问题的血小板聚集和凝血模型的动机，再次证明高阶收敛速度的移动域。
We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter δ, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains is handled through a combination of rapid node set modification, a new high-order semi-Lagrangian method that utilizes the new tuning-free overlapped RBF-FD method, and a high-order time-integration method. The resulting framework has no tuning parameters and has O (N log N) time complexity. We demonstrate high-orders of convergence for advection-diffusion equations on time-varying 2D and 3D domains for both small and large Peclet numbers. We also present timings that verify our complexity estimates. Finally, we utilize our method to solve a coupled 3D problem motivated by models of platelet aggregation and coagulation, once again demonstrating high-order convergence rates on a moving domain.