A Special Session in Computational Mathematics at the
AMS Central Section Meeting
September 14-15, University of Texas, San Antonio, San Antonio 2024


Session #SS25:


Advances in Mathematical and Numerical Analysis of Partial Differential Equations for Application-Oriented Computations


Organizers:


Abstract:

The latest advances in computational analysis for partial differential equations with realistic applications are the focus in this mini-session. We bring together investigators from various backgrounds to present and discuss the cutting-edge theory and methods of application-oriented computations for PDE. The latest machine learning challenges in these fields will also be explored. This mini-session serves as a platform to exchange new ideas, extend academic networks, and incubate future cooperation. Graduate students are encouraged to participate in the special event.


SS25 Informational Flyer: AMS FALL 2024 Central Sectional Meeting: SS25.


Important Deadlines

Meeting & Hotel Reservations: August 23, 2024



Instructions for Submission


Special Session SS25 Confirmed Speakers (as of May 22, 2024)

This list has no particular order until a future date.
  1. Time: Not specified
    Title: Comparison of the Spectral Method and the Continuous Interior Penalty Finite Element Method for the Phase Field Crystal Equation
    Presenter: Natasha Sharma
    Affiliation: University of Texas at El Paso
    Abstract: In this talk, we present two recently developed numerical schemes for the phase field crystal (PFC) equation and report on our findings based on a series of performance tests in terms of efficiency, accuracy, and ease of implementation. We focus our comparative study on formulations that originate from spectral methods and finite element methods. Numerical results will be presented and discussed. In addition, the dynamics of the physical processes captured by the PFC will be presented to illustrate the robustness of the proposed methods for modeling different applications such as crystal growth in a supercooled liquid and crack propagation in a ductile material.

  2. Time: Not specified
    Title: Efficient Krylov-Based Exponential Time Differencing Method
    Presenter: Bruce Wade
    Affiliation: University of Louisiana at Lafayette
    Abstract: We discuss a Krylov subspace approximation-based locally extrapolated exponential time differencing method for nonlinear advection-diffusion-reaction systems and compare its performance to established schemes.

  3. Time: Not specified
    Title: An Exploration of a Nonconventional Way of Operator Splitting Methods for Solving Diffusion Equations
    Presenter: Qin Sheng
    Affiliation: Baylor University
    Abstract: This talk concerns an application of an operator decomposition formula in Lagrangian optimizations for solving linear and semi-linear diffusion equations. It has revealed in our preliminary investigations that the nonconventional splitting configuration may not only provides an effective enhancement to conventional splitting methods, but also introduces a flexible way for constructing multi-parameter operator splitting strategies in many cutting-edge applications. Simulation experiments will be given.

  4. Time: Not specified
    Title: Integral Equation Method for the 1D Steady State Poisson-Nernst-Planck Equations
    Presenter: Robert Krasny
    Affiliation: University of Michigan
    Abstract: An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations describing charge transport through transmembrane ion channels. The differential equations are cast as integral equations using Green’s 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by midpoint and trapezoid rules and the resulting algebraic equations are solved by Gummel iteration. Numerical tests demonstrate the method’s 2nd order accuracy and ability to resolve sharp boundary layers. Results for a multi-domain Potassium ion channel model are in good agreement with published results. This is joint work with Zhen Chao (Western Washington University) and Weihua Geng (Southern Methodist University).

  5. Time: Not specified
    Title: Modeling Analytical Ultracentrifugation Experiments with an Adaptive Space-Time Finite Volume Solution of the Lamm Equation for Non-Ideal Solute and Solvent Conditions.
    Presenter: Weiming Cao
    Affiliation: University of Texas at San Anotonio
    Abstract: We present a solution of the Lamm equation based on the adaptive space-time finite volume method (ASTFVM) for modeling solutes displaying variable sedimentation and diffusion coefficients. Such conditions are observed when sedimenting molecules are measured at high concentration where they exhibit concentration-dependent nonideality or when co-sedimenting solutes create a density and viscosity gradient during the experiment. Variable sedimentation and diffusion coefficients are also observed in experiments performed in compressible solvents, where a density and viscosity gradient develops during rotor acceleration. The ASTFVM method provides a robust, accurate and efficient solution to these cases and has been integrated into the optimization methods in the UltraScan software for analytical ultracentrifugation analysis in biophysics and biochemistry.

  6. Time: Not specified
    Title: Convergence of Adaptive Least-Squares Finite Element Methods.
    Presenter: Jaeun Ku
    Affiliation: Oklahoma State University
    Abstract: In this talk, we consider adaptive procedures for least-squares finite element methods. We establish that the sequence of the approximation solutions generated by the adaptive procedure is a Cauchy sequence in a Banach space. This leads to the conclusion that the sequence converges. In order to force the sequence converging to the true solutions, we propose a refinement strategy using weighted least-squares functional as an a posteriori error indicator to identify the local regions to refine the current underlying mesh.

  7. Time: Not specified
    Title: A Fourth-Order Exponential Time Differencing Scheme with a Real and Distinct Poles Rational Approximation for Solving Nonlinear Systems of Reaction Diffusion Equations.
    Presenter: Wisdom K. Attipoe
    Affiliation: Clarkson University
    Abstract: Reaction-diffusion systems are mathematical models that describe the spatiotemporal dynamics of chemical substances as they diffuse and react. Variety of time discretization schemes have been developed to solve the stiff ODE system resulting from the spatial-discretization of the PDE. The quest to develop more efficient and accurate schemes to handle very stiff and non-smooth problems is ever increasing. In this work, we develop a fourth-order, L-stable, parallelizable exponential time differencing scheme (ETD) by approximating the matrix exponentials in the class of ETD Runge-Kutta schemes with a fourth-order non- Padé rational function having real distinct poles (RDP). A variety of non-linear reaction-diffusion systems having Dirichlet, Neumann and periodic boundary conditions are used to empirically validate the order of convergence of the scheme and compare its performance with existing fourth order schemes.

  8. Time: Not specified
    Title: Euler Maruyama Numerical Approximation of Stochastic Differential Equations with Continuously Distributed Delay.
    Presenter: Roshini Samanthi Gallage
    Affiliation: University of Oklahoma
    Abstract: In this talk, processes with continuously distributed delay which depend on weighted averages of past states over the entire time lag interval are discussed. We give generalized Khasminskii-type conditions which along with local Lipschitz conditions are sufficient to guarantee the existence of a unique solution of certain n-dimensional nonlinear stochastic delay differential equations (SDDEs) with continuously distributed delay. We give conditions under which Euler-Maruyama numerical approximations of such nonlinear SDDEs converge in probability to their exact solutions. Further, the Euler Maruyama numerical approximation results of some SDDE models are discussed to analyze the numerical approximation error values and trajectory behaviors. Joint work with Dr. Harry Randolph Hughes

  9. Time: Not specified
    Title: A Fourth-Order Exponential Time Differencing Scheme with Dimensional Splitting for Solving Semilinear Parabolic PDE.
    Presenter: Emmanuel O. Asante-Asamani
    Affiliation: Clarkson University
    Abstract: A fourth-order exponential time differencing (ETD) Runge-Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction-diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Padé (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20X speed-up in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions.

  10. Time: Not specified
    Title: Numerical Experiments Using the Barycentric Lagrange Treecode to Compute Correlated Random Displacements for Brownian Dynamics Simulations
    Presenter: Lei Wang
    Affiliation: University of Wisconsin- Milwaukee
    Abstract: To account for hydrodynamic interactions among solvated molecules, Brownian dynamics simulations require correlated random displacements ${\bf g} = D^{1/2}{\bf z}$, where $D$ is the $3N \times 3N$ Rotne-Prager-Yamakawa diffusion tensor for a system of $N$ particles and ${\bf z}$ is a standard normal random vector. The Spectral Lanczos Decomposition Method (SLDM) computes a sequence of Krylov subspace approximations ${\bf g}_k\,{\to}\,{\bf g}$, but each step requires a dense matrix-vector product $D{\bf q}$ with a Lanczos vector ${\bf q}$, and the $O(N^2)$ cost of computing the product by direct summation (DS) is an obstacle for large-scale simulations. This work employs the barycentric Lagrange treecode (BLTC) to reduce the cost of the matrix-vector product to $O(N\log N)$ while introducing a controllable approximation error. Numerical experiments compare the performance of SLDM-DS and SLDM-BLTC with particular attention on the effect of the BLTC approximation error.

  11. Time: Not specified
    Title: Exponential Time Differencing Method vs. Integrating Factor method in application to the multidimensional coupled nonlinear Schrödinger equations.
    Presenter: Harish Bhatt
    Affiliation: Utah Valley University
    Abstract: Coupled nonlinear Schrödinger equations (CNLSEs) are an extension of the nonlinear Schrödinger equation (NLSE) to model systems with multiple interacting waves, such as those in fiber optics and nonlinear optical media for laser technology. Despite their widespread applications, analytical solutions for most CNLSEs are either unknown or challenging to compute, necessitating numerical approaches. In our talk, we introduce the Runge-Kutta based Exponential Time Differencing (ETD) and integrating factor (IF) methods in combination with the Fourier spectral method, for simulating multi-dimensional CNLSEs. Numerical findings show that the ETD method outperforms the IF method in accuracy, efficiency, and preservation of mass and energy over long durations, while maintaining the expected order of accuracy.

  12. Time: Not specified
    Title: A look at approximating the degenerate Kawarada equation utilizing non-uniform finite differences.
    Presenter: Eduardo Servin Torres
    Affiliation: Baylor University
    Abstract: The solid fuel combustion process found in the oil industry can be modeled asymp- totically through non-linear Kawarada partial differential equations. While their non-linear terms capture features of the combustion momentum, they make required mathematical analysis ex- tremely challenging. It turns out that such a non-linear term is frozen or linearized for simplicity in various analysis. In this talk we will study an novel approach that does not require freezing any non-linear terms in the numerical stability, convergence, and other key ingredients of the finite difference solutions. The approach will be based on non-uniform meshes seeking robotically the quenching singularity. The pointwise convergence order will be estimated with the aid of Milne device. Finally, simulations will be given to verify our theoretical conclusions.

  13. Time: Not specified
    Title: A Finite Element Model for Fracture in a Transversely Isotropic Elastic Body.
    Presenter: Saugata Ghosh
    Affiliation: University of Texas Rio Grande Valley
    Abstract: This talk presents a mathematical description of the mode-I fracture boundary value problem in a relatively new class of elastic bodies. After introducing some basic notions about elasticity, a new nonlinear constitutive relation between stress and strain is derived. The balance of linear momentum yields a second-order quasilinear elliptic partial differential equation system. A well-posed boundary value problem (BVP) will be derived for the case of an elastic body containing a single plane-strain fracture. The BVP is discretized by a continuous Galerkin-type finite element method coupled with a Picard-type linearization. Finally, we present some interesting simulation results that show contrasts in the crack-tip strain singularity compared with the classical elasticity model. This is a joint project with Dr. Bhatta (University of Texas-Rio Grande Valley) and Dr. Mallikarjunaiah (Texas A&M University-Corpus Christi).

  14. Time: Not specified
    Title: Mathematical modeling of brittle fracture in the context of strain-limiting theories of elasticity: State of the art and future directions.
    Presenter: S. M. Mallikarjunaiah
    Affiliation: Texas A& M University-Corpus Christi
    Abstract: With their algebraically nonlinear constitutive relations between the Cauchy stress and the linearized strain, strain-limiting theories of elasticity are an attractive and promising framework for modeling brittle fracture. The most significant advantage of characterizing the material response in the context of such theories is that one can limit the strains to physically meaningful levels even when the stresses become arbitrarily large, a fact that should reassure and convince the research community. In this talk, I will present the state-of-the-art progress in developing mathematical and computational models to study the quasi-static and dynamic evolution of crack tips in elastic solids. I will also present some inherent challenges from this new school of thought.

  15. Time: Not specified
    Title: Soret effect on weakly nonlinear solutions for a convective flow in porous media.
    Dambaru Bhatta
    Affiliation: UTRGV
    Abstract: We investigate Soret effect due to thermo-solutal convection in a horizontal porous layer. The governing system consists of Darcy momentum equation, continuity equation, heat equation and solute transport equation. This convection occurs due to a buoyancy force caused by density variation because of temperature and solute concentration differences. We consider a no-flow and vertically varying basic state system, and then, using weakly nonlinear approach, various order perturbed systems are obtained and solved using normal mode process. An amplitude equation is also derived. Numerical results are presented to explore Soret effect.

  16. Time: Not specified
    Title: Decoupling schemes for multicontinuum problems in fractured porous media.
    Maria Vasilyeva
    Affiliation: Texas A& M University-Corpus Christi
    Abstract:We consider the coupled system of partial differential equations that describe flow in fractured porous media. Multicontinuum and multiscale approaches are used to describe such types of problems, where the permeability of each continuum is highly heterogeneous and has a significant difference. In this work, we introduce an effective decoupling method that separates the equations for each continuum. The approach is built on an implicit-explicit approximation by time with an explicit treatment of the coupling term. We have developed and analyzed continuum decoupling techniques for fine-scale and multiscale space approximations. Numerical results demonstrate that the proposed scheme is stable, accurate, and computationally efficient.


Contact

Primary email address: bruce.wade@louisiana.edu

UL Lafayette main contact:
Dr. Bruce Wade, Professor & Department Head
215 Maxim Doucet Hall
337.482.5172 or 337.482.5173
bruce.wade@louisiana.edu

Mailing address:
Department of Mathematics
P.O. Box 43568
University of Louisiana at Lafayette
Lafayette, LA 70504-3568

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