A well-balanced meshless tsunami propagation and inundation model

We present a novel meshless tsunami propagation and inundation model. We discretize the nonlinear shallow-water equations using a well-balanced scheme relying on radial basis function based finite differences. The inundation model relies on radial basis function generated extrapolation from the wet points closest to the wet-dry interface into the dry region. Numerical results against standard one- and two-dimensional benchmarks are presented.Comment: 20 pages, 13 figure

Meshfree Methods for PDEs on Surfaces

This dissertation focuses on meshfree methods for solving surface partial differential equations (PDEs). These PDEs arise in many areas of science and engineering where they are used to model phenomena ranging from atmospheric dynamics on earth to chemical signaling on cell membranes. Meshfree methods have been shown to be effective for solving surface PDEs and are attractive alternatives to mesh-based methods such as finite differences/elements since they do not require a mesh and can be used for surfaces represented only by a point cloud. The dissertation is subdivided into two papers and software. In the first paper, we examine the performance and accuracy of two popular meshfree methods for surface PDEs:generalized moving least squares (GMLS) and radial basis function-finite differences (RBF-FD). While these methods are computationally efficient and can give high orders of accuracy for smooth problems, there are no published works that have systematically compared their benefits and shortcomings. We perform such a comparison by examining their convergence rates for approximating the surface gradient, divergence, and Laplacian on the sphere and a torus as the resolution of the discretization increases. We investigate these convergence rates also as the various parameters of the methods are changed. We also compare the overall efficiencies of the methods in terms of accuracy per computation cost. The second paper is focused on developing a novel meshfree geometric multilevel (MGM) method for solving linear systems associated with meshfree discretizations of elliptic PDEs on surfaces represented by point clouds. Multilevel (or multigrid) methods are efficient iterative methods for solving linear systems that arise in numerical PDEs. The key components for multilevel methods: \grid coarsening, restriction/ interpolation operators coarsening, and smoothing. The first three components present challenges for meshfree methods since there are no grids or mesh structures, only point clouds. To overcome these challenges, we develop a geometric point cloud coarsening method based on Poisson disk sampling, interpolation/ restriction operators based on RBF-FD, and apply Galerkin projections to coarsen the operator. We test MGM as a standalone solver and preconditioner for Krylov subspace methods on various test problems using RBF-FD and GMLS discretizations, and numerically analyze convergence rates, scaling, and efficiency with increasing point cloud resolution. We finish with several application problems. We conclude the dissertation with a description of two new software packages. The first one is our MGM framework for solving elliptic surface PDEs. This package is built in Python and utilizes NumPy and SciPy for the data structures (arrays and sparse matrices), solvers (Krylov subspace methods, Sparse LU), and C++ for the smoothers and point cloud coarsening. The other package is the RBFToolkit which has a Python version and a C++ version. The latter uses the performance library Kokkos, which allows for the abstraction of parallelism and data management for shared memory computing architectures. The code utilizes OpenMP for CPU parallelism and can be extended to GPU architectures

Doctor of Philosophy

dissertationPlatelet aggregation, an important part of the development of blood clots, is a complex process involving both mechanical interaction between platelets and blood, and chemical transport on and o the surfaces of those platelets. Radial Basis Function (RBF) interpolation is a meshfree method for the interpolation of multidimensional scattered data, and therefore well-suited for the development of meshfree numerical methods. This dissertation explores the use of RBF interpolation for the simulation of both the chemistry and mechanics of platelet aggregation. We rst develop a parametric RBF representation for closed platelet surfaces represented by scattered nodes in both two and three dimensions. We compare this new RBF model to Fourier models in terms of computational cost and errors in shape representation. We then augment the Immersed Boundary (IB) method, a method for uid-structure interaction, with our RBF geometric model. We apply the resultant method to a simulation of platelet aggregation, and present comparisons against the traditional IB method. We next consider a two-dimensional problem where platelets are suspended in a stationary fluid, with chemical diusion in the fluid and chemical reaction-diusion on platelet surfaces. To tackle the latter, we propose a new method based on RBF-generated nite dierences (RBF-FD) for solving partial dierential equations (PDEs) on surfaces embedded in 2D domains. To robustly tackle the former, we remove a limitation of the Augmented Forcing method (AFM), a method for solving PDEs on domains containing curved objects, using RBF-based symmetric Hermite interpolation. Next, we extend our RBF-FD method to the numerical solution of PDEs on surfaces embedded in 3D domains, proposing a new method of stabilizing RBF-FD discretizations on surfaces. We perform convergence studies and present applications motivated by biology. We conclude with a summary of the thesis research and present an overview of future research directions, including spectrally-accurate projection methods, an extension of the Regularized Stokeslet method, RBF-FD for variable-coecient diusion, and boundary conditions for RBF-FD

Radial Basis Function Based Quadrature over Smooth Surfaces

The numerical approximation of denite integrals, or quadrature, often involves the construction of an interpolant of the integrand and subsequent integration of the interpolant. It is natural to rely on polynomial interpolants in the case ofone dimension; however, extension of integration of polynomial interpolants to two or more dimensions can be costly andunstable. A method for computing surface integrals on the sphere is detailed in the literature (Reeger and Fornberg,Studies in Applied Mathematics, 2016). The method uses local radial basis function (RBF) interpolation to reducecomputational complexity when generating quadrature weights for the particular node set. This thesis expands upon thesame spherical quadrature method and applies it to an arbitrary smooth closed surface dened by a set of quadraturenodes and triangulation

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in ℝd. The novelty of the method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds

A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on ℝ, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed