PB1: Contributed Session

Room: Old Main Academic Center 3030

Webex Link

Chair: Vu Thai Luan, Mississippi State University

Ray Treinen, Texas State University

Time: 2:40 pm - 3:00 pm (CST)

Title: Spectral methods for some capillary surfaces described by generating curves


We consider bounded capillary surfaces that are constructed by generating curves. This includes radically symmetric surfaces, as well as lower dimensional fluid-fluid interfaces. We use the arc-length representation of the differential equations for these surfaces to allow for vertical points and inflection points along the generating curve. These considerations admit capillary tubes, sensible drops, and fluids in annular tubes. We present a pseudo-spectral method for approximating solutions to the associated boundary value problems based on interpolation by Chebyshev polynomials. This method is observably more stable than the traditional shooting method and computationally leaner and fast. The algorithm is also adaptive, but does the adaptive automation in Chebfun.

Jorge Cisneros Paz,  University of Washington

Time:  3:00 pm - 3:20 pm (CST)

Title: Split-step methods with finite difference schemes and analytic continuation formulas


Finite difference schemes provide a popular and intuitive approach to numerically solve nonlinear initial-boundary value problems (IBVPs). Often, this leads to the introduction of ghost points, where the numerical method depends on grid points outside of the working domain. The usual heuristics of doing this for second-order problems do not generalize to higher order, and incorporating boundary conditions and addressing ghost points is a serious numerical issue. Our approach tackles this problem by the implementation of split-step methods to separately solve the linear and nonlinear problems. In this talk, I will present the the Unified Transform Method (UTM), introduced by A. S. Fokas, and how it is used to solve initial-boundary value problems for linear IBVPs. The UTM solution representations are then treated to give analytic continuation formulas that can be applied at ghost points in the split-step method. We present our developments via the application to the nonlinear Schrodinger equation on the finite interval. We discuss the continuum limit of the solutions and numerical results.

Nguyen Van Hoang, Mississippi State University

Time:  3:20 pm - 3:40 pm (CST)

Title: Efficient exponential methods for genetic regulatory systems


In recent years, gene regulatory networks (GRNs) have attracted a lot of interest as they play a crucial role in modeling the protein regulation processes. Among many numerical schemes which were introduced for simulating these systems, the classical fourth-order Runge-Kutta method (RK4) has been often used. When simulating GRNs, it is difficult to characterize some important properties of the system, such as stability and phase properties. Very recently, splitting methods, exponentially fitted two-derivative Runge- Kutta (EFTDRK) methods and classical exponential Runge-Kutta (ExpRK)-type methods have been also considered. In this work, we propose a class of partitioned ExpRK methods and exponential Rosenbrock methods for the simulation of genetic regulatory systems. Moreover, we also investigate the stability and phase properties of the proposed schemes. Finally, based on a set of benchmark test problems including two- and threegene systems, we demonstrate the accuracy and efficiency of the newly derived methods in comparison with several popular schemes.

This talk is based on a joint work with Vu Thai Luan and Julius Ehigie.

Trky Alhsmy, Mississippi State University 

Time:  3:40 pm - 4:00 pm (CST)

Title: High-order adaptive exponential Runge—Kutta methods


Exponential Runge-Kutta (ExpRK) methods have shown to be well-suited for the time discretization of stiff semilinear parabolic PDEs. The construction of stiffly-accurate ExpRK schemes requires solving a system of stiff order conditions which involve matrix functions. So far, methods up to order 5 have been derived by relaxing one or more order conditions (depending on a given order of accuracy). These schemes, however, allow using with constant stepsizes only. In this talk, we will derive new and efficient ExpRK schemes of high orders which not only fulfill the stiff order conditions in the strong sense and but also support variable step sizes implementation. Numerical examples are given to verify the accuracy and to illustrate the efficiency of the newly constructed ExpRK schemes.

This talk is based on a joint work with Vu Thai Luan.

Xuejian Li, Missouri University of Science and Technology

Time:  4:00 pm - 4:20 pm (CST)

Title: Incremental Proper Orthogonal Decomposition and Its Applications


Proper orthogonal decomposition (POD) is a data approximation method that aims at obtaining a low-dimensional representation of high-dimensional processes. It does so by creating an optimal lower order basis, called POD modes, to minimize the information loss. POD has been widely used in field of fluid dynamics, image process, signal analysis and data compression. The common skills to calculate POD challenges the computer resource when dealing with very largescale data. In this work, we present an incremental technique to address this issue. The algorithm is developed for both standard Rspace and weighted Rn space. We also provide error estimations to easily monitor the information loss along the incremental process. The applications are then given to producing dominant POD modes for reduced basis method and data compression for PDE-constrained optimization. Finally, the numerical examples are provided to validate the proposed method.