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: 9th Dec 2022, 12:50:11am CET

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Session Overview
MS 10: high order methods: design and analysis
Monday, 12/July/2021:
1:50pm - 3:50pm

Session Chair: Zheng Chen
Session Chair: Xinghui Zhong
Virtual location: Zoom 4

Session Abstract

High order methods such as discontinuous Galerkin methods, weighted essentially non-

oscillatory (WENO) methods and spectral methods have found rapid applications in such

diverse areas as aeroacoustics, electro-magnetism, gas dynamics, granular flows, magneto-

hydrodynamics, meteorology, modeling of shallow water, oceanography, oil recovery

simulation, semiconductor device simulation, transport of contaminant in porous media,

turbomachinery, turbulent flows, viscoelastic flows and weather forecasting, among many


This mini-symposium is designed as a forum for researchers to exchange ideas on recent

development on algorithm design, analysis and applications of high order methods, with

emphasis on discontinuous Galerkin methods, WENO methods and spectral methods.

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1:50pm - 2:20pm

Second Order in Time Bound-Preserving Implicit Pressure Explicit Concentration Methods for Contaminant Transportation in Porous Media

Yang Yang

Michigan Technological University, United States of America

In this talk, we apply the implicit pressure and explicit concentration (IMPEC) methods for compressible miscible displacements in porous media. The method can yield much larger time step size compared with the fully explicit method. However, most IMPEC methods are only of first order in time. In this talk, we will discuss how to construct a second order in time IMPEC method. The basic idea is to add the correction stage after each time step. Moreover, we will also construct the bound-preserving technique to preserve the upper and lower bounds of the concentration. Numerical experiments will be given to demonstrate the good performance of the proposed method.

2:20pm - 2:50pm

Superconvergence Reconstruction of Discontinuous Galerkin Methods: Efficiency and Flexibility

Xiaozhou Li

University of Electronic Science and Technology of China, China, People's Republic of

One of the beneficial properties of the higher Galerkin is the superconvergence properties. For example, the standard DG error has an accuracy order of 2k+1 in the negative norm (or, say, in a dual Sobolev space). With a specially designed reconstruction technique, namely, the Smoothness-Increasing Accuracy-Conserving (SIAC) filter, one can reproduce the superconvergence in the solution space (in the L2 norm). Previous studies have concentrated on the superconvergence order and neglected the vital role that the structure of the designed filter plays in the efficiency and flexibility of the reconstruction algorithm. In this talk, we investigate the structure of the filter plays during the reconstruction process. This study gives insight into the essential properties the filter must maintain and what structures people can adjust to create meet higher efficiency and flexibility requirements for practical applications. Further, we demonstrate that the ability of the SIAC filter to extract superconvergence is unaffected by the modification in both theoretical proof and numerical examples, with a more efficient and more flexible structure in the meanwhile.

2:50pm - 3:20pm

Analysis of a multiscale discontinuous Galerkin method for the stationary Schrodinger equation in 2D Simulation of Quantum Transport

Bo Dong, Wei Wang

University of Massachusetts Dartmouth, United States of America

We develop and analyze a multiscale discontinuous Galerkin (DG) method for two-dimensional stationary Schrodinger equations in quantum transport. The quantum directional coupler under consideration has frequency change mainly in one direction, so we use oscillatory non-polynomial basis functions in that direction and polynomial basis in the other direction. We observe that the method achieves at least second-order convergence on coarse meshes and optimal higher order convergence when the mesh size is refined to the scale of the wave length.

3:20pm - 3:50pm

Analysis of local discontinuous Galerkin methods with generalized numerical fluxes for linearized KdV equations

Xiong Meng, Jia Li, Dazhi Zhang, Boying Wu

Harbin Institute of Technology, China, People's Republic of

In this talk, we analyze the local discontinuous Galerkin method using generalized numerical fluxes for linearized

Korteweg--de Vries equations. The LDG scheme utilizes three independent weights in generalized numerical fluxes for convection

and dispersion terms so that it provides more flexibility. By using inherent relations of three different numerical viscosity coefficients resulting from numerical fluxes, we first show a uniform stability for the auxiliary variables and the prime variable as well as its time derivative. To derive optimal error estimates of LDG methods with generalized fluxes for high order wave equations, a suitable design of the numerical initial condition with optimal convergence properties for all variables is vital. To this end, we then define and analyze a special numerical initial condition, which is nothing but the LDG approximation with the same numerical fluxes to the corresponding steady--state equation. Thus, optimal error estimates of order $k+1$ are obtained by using an energy analysis in combination with some particularly designed generalized Gauss--Radau projections, where $k$ is the highest polynomial degree of discontinuous finite element space. Long time simulations of different numerical fluxes in the LDG scheme are also discussed. Numerical experiments are given to demonstrate the theoretical results.

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