Title: Solutions of Generalized KdV type Equations with Various Limits
Abstract: In this talk, we will consider the generalized Korteweg-de Vries (KdV) equation $$u_t+au_x+2buu_x+cu_{xxx}-du_{xx}=0$$ involving four parameters a, b, c, and d in a periodic domain. This equation has many physical applications including modeling the propagation of water waves. We will present some of the analytical and numerical results in various limiting cases of the parameters.

Title: A Multivariate Spline based Collocation Method for Numerical Solution of Partial Differential Equations
Abstract: We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to the second order elliptic PDE in non-divergence form. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings.

Title: A Symmetric Banded Matrix Isospectral Flow on Real Space
Abstract: In this talk, we discuss an isospectral flow of the form $\displaystyl \dot{P} = [[P’, P]_{du}, P]$ in the space of real matrices, where, $\dot{P}$ is the time derivative of matrix $P$, $P’$ is the transpose of matrix $P$, and $[P’, P]_{du}$ is the same as matrix $[P’, P]$ on the upper triangular elements and all elements below the diagonal are zero. We prove that if the initial condition $P_0$ is a banded matrix having lower bandwidth $p$ with a simple and real spectrum, then $P(t)$ converges as $t \to \infty$ to a banded symmetric matrix having bandwidth $p$, isospectral to $P_0$. Also, the limit point has the same sign pattern in the $p^{th}$ subdiagonal elements as in $P_0$. We provide simulation results to highlight the transient behavior of $p^{th}$ subdiagonal elements under the flow.

Title: A parameter-free enriched Galerkin method for the Stokes equations
Abstract: We propose a parameter-free enriched Galerkin (EG) method for the Stokes equations using modified weak Galerkin (mWG) bilinear forms. The discrete inf-sup stability condition is a fundamental property of numerical methods for the Stokes equations, and it has been successfully achieved by recent work on a new EG method whose numerical velocity is a discontinuous piecewise linear function. In the EG method, interior penalty discontinuous Galerkin (IPDG) approaches have been adopted to treat the discontinuity of numerical velocity, and the approaches require a sufficiently large penalty parameter. In this work, for the same numerical velocity, we use mWG bilinear forms compatible with discontinuous piecewise polynomials without the penalty parameter. Therefore, our EG method is a parameter-free EG method for the Stokes equations. We prove that our parameter-free EG method guarantees the optimal convergence rates, and pressure-robustness is also achieved by employing a velocity reconstruction operator on the load vector of the right-hand side. Furthermore, the theoretical results are verified by various numerical examples.