Title: Pressure Robust Scheme for Incompressible Flow
Abstract: In this talk, we shall introduce the recent development regarding the pressure robust finite element method (FEM) for solving incompressible flow. We shall take weak Galerkin (WG) scheme as the example to demonstrate the proposed enhancement technique in designing the robust numerical schemes and then illustrate the extension to other finite element methods. Weak Galerkin (WG) Method is a natural extension of the classical Galerkin finite element method with advantages in many aspects. For example, due to its high structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations on the general meshing by providing the needed stability and accuracy. Due to the viscosity and pressure independence in the velocity approximation, our scheme is robust with small viscosity and/or large permeability, which tackles the crucial computational challenges in fluid simulation. We shall discuss the details in the implementation and theoretical analysis. Several numerical experiments will be tested to validate the theoretical conclusion.

Title: Convergence Analysis of Non-Monotone Finite Difference Methods for Approximating Viscosity Solutions of Stationary Hamilton-Jacobi Equations
Abstract: A new non-monotone finite difference (FD) approximation method is proposed for stationary Hamilton-Jacobi problems with Dirichlet boundary condition. The FD method has local truncation errors that are above the first order Godunov barrier for monotone methods, and it is proved to converge to the unique viscosity solution of the underlying first order fully nonlinear partial differential equation. A stabilization term called a numerical moment is used to ensure the proposed scheme is admissible, stable, and convergent. Numerical tests are provided that compare the accuracy of the proposed scheme to that of the Lax-Friedrich’s method.

Title: A New Unstaggered Locally Divergence-Free Finite Volume Scheme for Ideal and Shallow Water Magnetohydrodynamics
Abstract: Many mathematical models of fundamental interest are constrained by the infamous divergence-free condition of magnetic fields or velocity profiles. Seen within a wide range of astrophysical, geophysical, and engineering applications, such divergence-free constraints are physically and analytically exact. On a discrete level, however, an improper treatment of an identically-zero divergence may lead to large instabilities in solutions – even when applying methods that already successfully simulate fluids without this constraint. Thus, a careful algorithmic construction is required to ensure these constraints are exactly preserved within numerical approximations. In this talk, we restrict our attention to an important sub-class of divergence-free systems – the magnetohydrodynamic equations, keeping in mind that the developed method can be extended to other models constrained by zero-divergence. We present a new method that exactly preserves the divergence-free condition of the magnetohydrodynamic system on an unstaggered mesh. The designed method has been successfully tested on several examples.

Title: Localized method of particular solutions using polynomial basis functions for solving two-dimensional nonlinear partial differential equations.
Abstract: The localized method is one of the popular approaches in solving large-scale problems in science and engineering. In this paper, we implement the localized method of particular solutions using polynomial basis functions for solving various nonlinear problems. To validate our proposed numerical method, we present four numerical examples in regular and irregular domains which are solved by using localized method of particular solution with polynomial basis functions. We compared our numerical method with localized method of particular solutions using multiquadric radial basis function and numerical results clearly show that our numerical method is highly accurate, efficient, and outperformed the method using multiquadric radial basis function.