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.
Meshless solutions of PDE inverse problems on irregular geometries
School of Engineering and Applied Sciences, Harvard University, Cambridge MA 02138（工程与应用科学学院，哈佛大学，康科德马萨诸塞州02138）；School of Engineering（工程学院）；Applied Sciences, Harvard University, Cambridge MA 02138（应用科学，哈佛大学，康科德马萨诸塞州02138）；Department of Physics, Harvard University, Cambridge MA 02138（物理系，哈佛大学，康科德马萨诸塞州02138）
A novel Hermite RBF-based differential quadrature method for solving two-dimensional variable-order time fractional advection-diffusion equation with Neumann boundary condition
Jiangsu Normal University（江苏师范大学）；De Montfort University（德蒙福特大学）
We present a second-order monolithic method for solving incompressible Navier--Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and pressure variables are located at cell centers. Compact finite difference methods with ghost values are used to discretize the advection and diffusion terms of the velocity. A pressure gradient and divergence operator on the quadtree that use compact stencils are developed. Furthermore, the proposed method is extended to cubical domains with octree grids. Num…
Local meshless methods using RBFs augmented with monomials have become increasingly popular, due to the fact that they can be used to solve PDEs on scattered node sets in a dimension-independent way, with the ability to easily control the order of the method, but at a greater cost to execution time. We analyze this ability on a Poisson problem with mixed boundary conditions in 1D, 2D and 3D, and reproduce theoretical convergence orders practically, also in a dimension-independent manner, as demonstrated with a solution of Poisson's equation in an irregular 4D domain. The results are further co…
A new high-order finite-volume advection scheme on spherical Voronoi grids and a comparative study in a mimetic finite-volume moist shallow-water model
Institute of Mathematical and Computer Sciences, University of São Paulo（圣保罗大学数学与计算机科学学院）；Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学数学、统计与计算机科学学院）
Radial Basis Function-generated Finite Differences (RBF-FD) is a popular variant of local strong-form meshless methods that do not require a predefined connection between the nodes, making it easier to adapt node-distribution to the problem under consideration. This paper investigates an RBF-FD solution of time-domain acoustic wave propagation in the context of seismic modeling in the Earth's subsurface. Through a number of numerical tests, ranging from homogeneous to highly-heterogeneous velocity models including non-smooth irregular topography, we demonstrate that the present approach can be…
A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University（河海大学力学与材料学院水文水资源与水利工程国家重点实验室）；College of Mechanics and Materials, Hohai University（河海大学力学与材料学院）；Department of Geological Sciences, University of Alabama（阿拉巴马大学地质科学系）；Beijing Computational Science Research Center（北京计算科学研究中心）
A higher order pressure-stabilized virtual element formulation for the Stokes-Poisson-Boltzmann equations
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Indian Institute of Space Science and Technology（印度空间科学与技术学院）；Istituto di Matematica Applicata e Tecnologie Informatiche Consiglio Nazionale delle Ricerche（意大利国家研究委员会应用数学与信息技术研究所）
