COMPUTATIONAL METHODS WITH SPECTRAL-LIKE ACCURACY: MODELING, ALGORITHMS, AND SIMULATIONS

This dissertation focuses on the development of new mathematical models and numerical schemes for fluid dynamics problems, as well as novel algorithms for solving general partial differential equations (PDEs).

In Chapters 2 and 3, we present new models and algorithms for simulating multiphase fluid flows under the influence of multiphysical fields, including electric fields and general potential fields. These models are formulated using the phase field method, grounded in fundamental conservation laws and thermodynamic principles. We derive the governing equations and propose efficient numerical schemes for their simulation. The theoretical properties of these new models are analyzed and compared with those of classical models. A series of numerical experiments are provided to demonstrate the effectiveness and accuracy of the proposed methods.

Chapter 4 addresses the simulation of incompressible fluid flows with outflow boundary conditions. We introduce appropriate boundary conditions to properly handle outflow regions and develop unconditionally energy-stable numerical schemes. These schemes are based on a velocity-correction approach to decouple the velocity and pressure fields in the Navier–Stokes equations and incorporate the scalar auxiliary variable (SAV) method to ensure unconditional energy stability. Rigorous proofs of discrete energy stability are pro- vided, and comprehensive numerical experiments are conducted to validate the performance of the proposed methods in comparison with existing approaches.

In Chapter 5, we introduce a novel numerical framework called Functionally Connected Elements (FCE) for solving boundary and initial value problems. This approach constructs general piecewise-defined functions over partitioned domains that satisfy intrinsic C0 and C1 continuity across subdomain interfaces. We further combine the FCE framework with a least squares collocation method to solve a range of linear and nonlinear problems in one and two dimensions. Numerical results are presented to illustrate the accuracy and versatility of the proposed method.

Chapter 6 explores two new neural network-based frameworks for solving PDEs. The first framework is based on Extreme Learning Machines (ELM), for which we propose a novel parameter estimation technique. Numerical experiments show that this approach outperforms traditional methods in both accuracy and efficiency. The second framework, termed the Discontinuous Galerkin Neural Network (DGNN) method, introduces a neural network formulation capable of approximating functions with discontinuities. By coupling this formulation with a collocation-based least-squares method, we develop an efficient solver for hyperbolicconservation laws. Numerical results demonstrate that the DGNN method effectively captures discontinuous features in the solution.