# NEW NUMERICAL APPROXIMATIONS OF GEOLOGICAL PROCESSES IN HETEROGENEOUS SYSTEMS USING RADIAL BASIS FUNCTIONS

2021

## Document Type

Open Access Dissertation

## Degree Name

Doctor of Philosophy in Mathematical Sciences (PhD)

Department of Mathematical Sciences

Cecile Piret

John S. Gierke

## Committee Member 2

Alexander E. Labovsky

Jiguang Sun

## Abstract

This dissertation includes four chapters. A brief description of each chapter is organized as follows. The first chapter provides an introduction to the RBF method. The chapter follows the historical progression of the Radial Basis Function (RBF) method while outlining the method’s advantages and disadvantages. A brief introduction about RBF interpolation, the RBF-FD method, and how to use it to solve PDEs is provided. Chapter 2 introduces a novel computationally efficient RBF-FD algorithm to solve the groundwater flow equation in the presence of an active well. We show that our method analytically handles the singularities in the PDE caused by the active well and can also solve the equation in complex heterogeneous environments. Furthermore, the numerical results from our method were compared against the United States Geological Survey MODFLOW a finite-difference, based groundwater flow model. Chapter 3 focuses on solving PDEs in the presence of an interface with discontinuous hydraulic conductivity values. We introduce a robust RBF-FD methodology to handle discontinuities of the PDE across any interface. The method is then numerically tested on both line and curved interfaces, and numerical results are presented. We show that the proposed algorithm can achieve high algebraic order convergence. Chapter 4 is a study of the performance of using the RBF–FD method with the parallel ODE time solver Parareal algorithm. RBF–FD method is only used to discretize a PDE in space, and therefore, choosing an appropriate time–solver that matches the RBF method is an important task. The differentiation matrices produced with the RBF–FD method sometimes contain spurious eigenvalues, which could cause us to choose unacceptably small-time steps. Chapter 4 introduces a robust approach to use the RBF-FD method and the Parareal algorithm to address this issue.

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