Abstract : Many problems in science and engineering involve multi-physical processes where their complex interactions occur at a wide range of spatial and temporal scales. They can be modeled by a set of nonlinear PDEs, each representing different physical phenomena. Numerical solutions of such problems remain a quite challenging task due to the presence of multiple spatial and time scales. The goals of this minisymposium are first to present some of latest developments in numerical methods for such complex PDEs, and second to bring together experts, young researchers, and students working in this field to exchange ideas and to initiate collaborations.
Organizer(s) : Vu Thai Luan, Amanda Diegel, Aaron Rapp
[01855] Accelerating nonlinear solvers with continuous data assimilation
Format : Talk at Waseda University
Author(s) :
Leo Rebholz (Clemson University)
Abstract : We show how continuous data assimilation can be used to accelerate convergence in nonlinear solvers for steady PDE. We prove that for incompressible flow problems, with sufficient measurement data the linear convergence rate of Picard iterations can be improved. Numerical tests illustrate the theory.
[01884] Modified exponential Rosenbrock methods to increase their accuracy
Format : Online Talk on Zoom
Author(s) :
Begoña Cano (Universidad de Valladolid)
María Jesús Moreta (Universidad Complutense de Madrid)
Abstract : In this talk a technique will be described to avoid order reduction when integrating nonlinear initial boundary value
problems with exponential Rosenbrock methods. The technique does not require to impose any stiff order conditions
but to add some terms related to the information on the boundary. Theoretical results on local and global error will be
given as well as some numerical comparisons.
[01971] Low Regularity Integrators for Semilinear Parabolic Equations with Maximum Bound Principles
Format : Talk at Waseda University
Author(s) :
Cao-Kha Doan (Auburn University)
Lili Ju (University of South Carolina)
Thi-Thao-Phuong Hoang (Auburn University)
Katharina Schratz (Sorbonne Université)
Abstract : This work is concerned with structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen-Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel’s formula, first and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, some semi-discrete and fully-discrete error estimates are also successfully derived under the low regularity requirement that the corresponding exact solution is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes can be more accurate and achieve better convergence than classic exponential time differencing (ETD) schemes, especially when the interfacial parameter approaches zero.
[01739] Unconditionally stable numerical methods for Cahn-Hilliard-Navier-Stokes-Darcy system with different densities and viscosities
Format : Online Talk on Zoom
Author(s) :
Xiaoming He (Missouri University of Science and Technology)
Yali Gao (Northwestern Polytechnical University)
Daozhi Han (University at Buffalo)
Ulrich Rüde (Friedrich-Alexander-University of Erlangen-Nuremberg)
Abstract : In this presentation, we consider the numerical modeling and simulation via the phase field approach for coupled two-phase free flow and two-phase porous media flow of different densities and viscosities. The model consists of the Cahn-Hilliard-Navier-Stokes equations in the free flow region and the Cahn-Hilliard-Darcy equations in porous media that are coupled by several domain interface conditions. It is showed that the coupled model satisfies an energy law. Then we first propose a coupled unconditionally stable finite element method for solving this model and analyze the energy stability for this method. Furthermore, based on the ideas of pressure stabilization and artificial compressibility, we propose an unconditionally stable time stepping method that decouples the computation of the phase field variable, the velocity and pressure of free flow, the velocity and pressure of porous media, hence significantly reduces the computational cost. The energy stability of this decoupled scheme with the finite element spatial discretization is rigorously established. We verify numerically that our schemes are convergent and energy-law preserving. Numerical experiments are also performed to illustrate the features of two-phase flows in the coupled free flow and porous media setting.
Abstract : Operator splitting is the dirty little thing we all do when a differential equation is too hard to solve monolithically. Many questions abound, however, regarding how to split a given problem, and many observations and "common knowledge" lack a theoretical understanding. In this talk, I take you on a tour through some of the practical aspects of operator splitting and while touching on some of the folklore that exists around it.
[02294] Adaptive exponential Runge--Kutta methods for stiff PDEs
Format : Talk at Waseda University
Author(s) :
Luan Vu Thai (Mississippi State University)
Abstract : Exponential Runge-Kutta methods have shown to be well-suited for stiff parabolic PDEs. Their constructions require solving stiff order conditions which involve matrix functions. Current schemes allow using with constant stepsizes only. In this talk, we will derive new schemes of high order which not only fulfill the stiff order conditions in the strong sense and but also support variable step sizes implementation. Numerical experiments are given to illustrate accuracy and efficiency of the new schemes.
[02660] The Dual-Wind Discontinuous Galerkin Method for Hamilton-Jacobi Equations
Format : Talk at Waseda University
Author(s) :
Aaron Rapp (University of the Virgin Islands)
Abstract : A discontinuous Galerkin (DG) finite-element interior calculus is used as a common framework to describe various DG approximation methods for second-order elliptic problems. This framework allows for the approximation of both primal and variational forms of second order differential equations. In this presentation, we will study the error from using the dual-wind DG derivatives to approximate the the solution to stationary and time-dependent Hamilton-Jacobi equations. Some analytical results will be presented, along with numerical examples that verify these results.
[05622] Well-balanced positivity-preserving DG methods for Euler equations with gravitation
Format : Online Talk on Zoom
Author(s) :
Jie Du (Tsinghua University)
Yang Yang (Michigan Technological University)
Fangyao Zhu (Michigan Technological University)
Abstract : We will look at high order discontinuous Galerkin methods with Lax-Friedrich fluxes for Euler equations under gravitational fields. A well-balanced (WB) positivity-preserving (PP) scheme should be constructed to solve the problem. We reformulate the source term such that it balances with the flux at the steady state. To obtain positive numerical density and pressure, we choose a special penalty coefficient in the Lax-Friedrich flux. Numerical experiments will be shown for the performance of the method.
