Solving partial differential equations (PDEs) can require numerical methods, especially for non-linear problems and complex geometry. Common numerical methods used today are the finite difference method (FDM), finite element method (FEM) and the finite volume method (FVM). These methods require a mesh or grid before a solution is attempted. Developing the mesh can require expensive preprocessing time and the quality of the mesh can have major effects on the solution. In recent years, meshless methods have become a research interest due to the simplicity of using scattered data points. Many types of meshless methods exist stemming from the spectral or pseudo-spectral methods, but the focus of this research involves a meshless method using radial basis function (RBF) interpolation. Radial basis functions (RBF) interpolation is a class of meshless method and can be used in solving partial differential equations. Radial basis functions are impressive because of the capability of multivariate interpolation over scattered data, even for data with discontinuities. Also, radial basis function interpolation is capable of spectral accuracy and exponential convergence. For infinitely smooth radial basis functions such as the Hardy Multiquadric and inverse Multiquadric, the RBF is dependent on a shape parameter that must be chosen properly to obtain accurate approximations. The optimum shape parameter can vary depending on the smoothness of the field. Typically, the shape parameter is chosen to be a large value rendering the RBF flat and yielding high condition number interpolation matrix. This strategy works well for smooth data and as shown to produce phenomenal results for problems in heat transfer and incompressible fluid dynamics. The approach of flat RBF or high condition matrices tends to fail for steep gradients and shocks. Instead, a low-value shape parameter rendering the RBF steep and the condition number of the interpolation matrix small should be used in the presence of steep gradients or shocks. This work demonstrates a method to capture steep gradients and shocks using a blended RBF approach. The method switches between flat and steep RBF interpolation depending on the smoothness of the data. Flat RBF or high condition number RBF interpolation is used for smooth regions maintaining high accuracy. Steep RBF or low condition number RBF interpolation provides stability for steep gradients and shocks. This method is demonstrated using several numerical experiments such as 1-D advection equation, 2-D advection equation, Burgers equation, 2-D inviscid compressible Euler equations, and the Navier-Stokes equations.
Doctor of Philosophy (Ph.D.)
College of Engineering and Computer Science
Mechanical and Aerospace Engineering
Length of Campus-only Access
Doctoral Dissertation (Open Access)
Harris, Michael, "A Localized Blended RBF Collocation Method for Effective Shock Capturing" (2018). Electronic Theses and Dissertations, 2004-2019. 6168.