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 $\delta$, 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.

翻译：我们提出了一个高顺序的辐射基函数(RBF-FD)框架,用于解决时间变化域的倒压-扩散方程式。我们的框架基于对最近开发的超压RBF-FD方法的概括化,该方法使用一种新型自动程序,用于在时间变化中心周围的变小区域对电压-FD加权计算RBF-FD加权数。该程序消除了重迭参数$delta$,从而使得RBF-FD移动域的变异矩阵能够不调整地组合。此外,我们的框架利用了一个简单有效的程序,更新移动域的变异矩阵。最后,在时间变化域中,通过快速节位修改、新的高调半Lagrangia方法,使用新的无调的RBFF-FD重叠法,以及高调调调调调的移动域的变异异矩阵。因此,我们的框架没有调整参数,而且已经用大量O(ND)时间变化基值的移动高比率。我们用一个高调的平流的平流计算方法来计算我们目前3的平流的平流的平流的平流的平流的平流、最后的平流的平流的平序。