Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 8th Dec 2022, 11:27:49pm CET

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Session Overview
MS 5: high order structure preserving numerical methods and applications
Wednesday, 14/July/2021:
2:00pm - 4:00pm

Session Chair: Yan Jiang
Virtual location: Zoom 3

Session Abstract

Mathematical models in the form of partial differential equations are extremely useful tools

in mathematical, scientific, and engineering communities. Development of robust, efficient

and highly accurate numerical algorithms for simulation of their solutions continues to be a

challenging task. High order numerical methods, such as discontinuous Galerkin method and

schemes with weighted essentially non-oscillatory (WENO) reconstructions have been under

great development for hyperbolic type PDEs with a broad range of applications in the past

few decades. One important, yet challenging, direction on further development of these high

order methods are to ensure the structure preserving properties, i.e. to develop high order

numerical methods that preserve certain structures or other fundamental continuum properties

of the underlying models exactly. Examples of structure preserving properties include (a)

asymptotic preserving that preserves the asymptotic limits of multi-scale PDE models, (b)

bound preserving that preserves the maximum principle or positivity of PDE solutions, (c)

energy preserving that preserves the energy exactly or energy dissipation at the semi discrete

or fully discrete level, (d) entropy stability, (e) free-stream preserving and (f) divergence free

methods that preserving the divergence free properties of fields. This mini-symposium aims

to bring together researchers to exchange ideas on recent development of high order methods

with emphasis on structure preserving techniques and applications.Proposed list of speakers

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2:00pm - 2:30pm

Asymptotic and positivity preserving methods for Kerr-Debye model with Lorentz dispersion

Zhichao Peng1,2, Vrushali A. Bokil3, Yingda Cheng2, Fengyan Li1

1Rensselaer Polytechnic Institute, United States of America; 2Michigan State University, United States of America; 3Oregon State University,United States of America

In this work, we consider the one-dimensional Kerr-Debye-Lorentz system to model the electromagnetic wave propagation in some nonlinear optical media. The nonlinearity in the polarization is a relaxed cubic Kerr type effect, and the polarization also includes the linear Lorentz dispersion. As the relaxation time goes to zero, the system will approach the Kerr-Lorentz model.

The objective here is to devise and analyze asymptotic preserving (AP) and positivity preserving (PP) methods for the Kerr-Debye-Lorentz model. Being AP, the methods address the stiffness of the model associated with small relaxation time, while capturing the correct Kerr-Lorentz limit on under-resolved meshes. Being PP, the third-order nonlinear susceptibility will stay non-negative and this is important for the energy stability. In the proposed methods, the nodal discontinuous Galerkin (DG) discretizations of arbitrary order accuracy are applied in space to effectively handle nonlinearity; in time, several first and second order methods are developed. The first order in time fully discrete schemes can be proved to be AP, PP and also energy stable. For the second order temporal accuracy, a novel modified exponential time integrator is proposed for the stiff part of the auxiliary differential equations modeling the electric polarization, and this is a key ingredient for the methods to be both AP and PP. In addition to a straightforward discretization of the constitutive law, we further propose a non-trivial energy-based approximation, with which the energy stability is also established mathematically. The performance of the methods are demonstrated by numerical examples.

2:30pm - 3:00pm

High order conservative sign-preserving time integrations and applications


Tsinghua University, China, People's Republic of

In this talk, we develop third-order conservative sign-preserving and steady-state-preserving time integrations and seek their applications. In multispecies and multireaction chemical reactive flows, the density and pressure are nonnegative, and the mass fraction for each species should be between 0 and 1. There are four main difficulties in constructing high-order bound-preserving techniques for multispecies and multireaction detonations. First of all, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Secondly, the mass fraction does not satisfy a maximum- principle and hence it is not easy to preserve the upper bound 1. Thirdly, in most of the previous works for gaseous denotation, the algorithm relies on second-order Strang splitting methods where the flux and stiff source terms can be solved separately, and the extension to high-order time discretization seems to be complicated. Finally, most of the previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this work, we will construct third-order conservative sign-preserving Runge-Kutta and multistep methods to overcome all these difficulties. The time integrations do not depend on the Strang splitting, i.e. we do not split the flux and the stiff source terms. Moreover, the time discretization can handle the stiff source with a large time step and preserves the steady-state. We also apply this time discretization to the classical Allen-Cahn equations and investigate the maximum- principle-preserving techniques.

3:00pm - 3:30pm

On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity

Yulong Xing

The Ohio State University, United States of America

Many partial differential equations (PDEs) can be written as a multi-symplectic Hamiltonian system, which obeys the multi-symplectic conservation law. In this presentation, we present and study semi-discrete discontinuous Galerkin (DG) methods for one-dimensional multi-symplectic Hamiltonian PDEs, and show that the proposed DG methods can simultaneously preserve the multi-symplectic structure and energy conservation with a general class of numerical fluxes, which includes the well-known central and alternating fluxes. Applications to the wave equation, the Benjamin--Bona--Mahony equation, the Camassa--Holm equation, the Korteweg--de Vries equation and the nonlinear Schrodinger equation are discussed. Some numerical results are provided to demonstrate the accuracy and long time behavior of the proposed methods.

3:30pm - 4:00pm

Recent Progress on Q^k Spectral Element Method: Accuracy, Monotonicity and Applications

Xiangxiong Zhang

Purdue University, United States of America

In the literature, spectral element methods usually refer to finite element methods with high order polynomial basis. The Q^k spectral element method has been a popular high order method (e.g., for solving wave equations) for more than three decades, obtained by continuous finite element method with tenor product polynomial of degree k and with at least (k+1)-point Gauss-Lobatto quadrature. In this talk, I will present some brand new results of this classical scheme, including its accuracy, monotonicity (stability), and examples of using monotonicity to construct high order bound-preserving schemes in various applications.

When the least accurate (k+1)-point Gauss-Lobatto quadrature is used, the spectral element method is also a finite difference (FD) scheme, and this FD scheme can sometimes be (k+2)-th order accurate for k>=2. We have proven this in error estimates for linear elliptic, parabolic, wave and Schrödinger equations.

We have proven that the fourth order accurate FD scheme (Q^2 spectral element method) is a product of two M-matrices thus monotone for a variable coefficient diffusion operator: this is the first time that a high order accurate scheme is proven monotone for a variable coefficient operator. We have also proven the fifth order accurate FD scheme (Q^3 spectral element method) is a product of three M-matrices thus monotone for the Poisson equation: this is the first time that a fifth order accurate discrete Laplacian is proven monotone in two dimensions (all previously known high order monotone discrete Laplacian in 2D are fourth order accurate).

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