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Title: Final Technical Report: High Order Discontinuous Galerkin Method and Applications

Abstract

This project aims at the developments and improvements of high order accurate discontinuous Galerkin finite element methods for solving partial differential equations arising from DOE applications. Algorithm development, improvement, analysis, implementation and applications aspects are addressed. The partial differential equations under study involve solutions with shocks or other discontinuities, rapidly changing regions, or multiscale solutions. These include the so-called convection dominated cases, such as the hyperbolic equations (e.g. Euler equations of gas dynamics), parabolic equations with small viscosities (e.g. high Reynolds number Navier-Stokes equations), KdV type equations with small dispersion describing water waves (small third derivative terms), time-dependent bi-harmonic equations and equations with hyper-viscosity (small fourth or higher even derivative terms), problems with highly oscillatory coefficients, or problems for which the macro-scale PDE models may not be correct in localized regions and a micro-scale model must be used instead. Furthermore, we are interested in the long time solutions of such equations with waves traveling a long distance, such as the electromagnetic waves. The requirement on numerical methods is thus high order accuracy, low dissipation/dispersion errors over long time, and a clean and sharp discontinuity resolution. It would also be highly desirable to have the numerical methods to be flexible formore » complicated geometry and boundary conditions, and to be easy in an adaptive and massively parallel environment. The discontinuous Galerkin finite element method has this potential, however the design and analysis of the discontinuous Galerkin method for each specific class of partial differential equations would require detailed research. In this project we have addressed the following crucial algorithm issues: (1) design and analysis for stable and accurate discontinuous Galerkin methods for various linear and nonlinear partial differential equations from applications, including surface diffusion and Willmore flow of graphs, non-equilibrium flow equations, Boltzmann-Poisson systems in nano devices, Vlasov-Poisson system in plasma physics, compressible multi-medium flows, shallow water equations, equations with delta-singularities in flocking and other models, Schr\"{o}dinger equations, Helmholtz equations, Maxwell's equations in Drude metamaterials, Cahn-Hilliard equation, gradient flow problems with interaction potentials, kinetic models of self-organized dynamics, and incompressible flows; (2) study of robustness and accuracy issues including well balanced schemes, positivity-preserving and general bound-preserving techniques, and nonlinearly stable high order weighted essentially non-oscillatory (WENO) limiters to control spurious oscillations; (3) study of superconvergence of discontinuous Galerkin for linear and nonlinear partial differential equations, which provides important tools to guide adaptive computation to save computational costs; (4) study of efficient, stable and high order accurate time discretizations including implicit-explicit (IMEX) time discretizations, stable Lax-Wendroff type time discretizations. There are 66 refereed journal papers published, which have cited partial support from this DOE grant. The algorithms developed have found wide spread applications including applications by DOE lab scientists.« less

Authors:
Publication Date:
Research Org.:
Chi-Wang Shu, Brown University
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1499046
Report Number(s):
DOE-BROWN-ER25863-1
DOE Contract Number:  
FG02-08ER25863
Resource Type:
Technical Report
Resource Relation:
Related Information: published journal papers citing support by this DOE grant
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; high order schemes; discontinuous Galerkin methods; partial differential equations; stability; accuracy; essentially non-oscillatory; robustness; bound preserving; positivity preserving

Citation Formats

Shu, Chi-Wang. Final Technical Report: High Order Discontinuous Galerkin Method and Applications. United States: N. p., 2019. Web. doi:10.2172/1499046.
Shu, Chi-Wang. Final Technical Report: High Order Discontinuous Galerkin Method and Applications. United States. doi:10.2172/1499046.
Shu, Chi-Wang. Tue . "Final Technical Report: High Order Discontinuous Galerkin Method and Applications". United States. doi:10.2172/1499046. https://www.osti.gov/servlets/purl/1499046.
@article{osti_1499046,
title = {Final Technical Report: High Order Discontinuous Galerkin Method and Applications},
author = {Shu, Chi-Wang},
abstractNote = {This project aims at the developments and improvements of high order accurate discontinuous Galerkin finite element methods for solving partial differential equations arising from DOE applications. Algorithm development, improvement, analysis, implementation and applications aspects are addressed. The partial differential equations under study involve solutions with shocks or other discontinuities, rapidly changing regions, or multiscale solutions. These include the so-called convection dominated cases, such as the hyperbolic equations (e.g. Euler equations of gas dynamics), parabolic equations with small viscosities (e.g. high Reynolds number Navier-Stokes equations), KdV type equations with small dispersion describing water waves (small third derivative terms), time-dependent bi-harmonic equations and equations with hyper-viscosity (small fourth or higher even derivative terms), problems with highly oscillatory coefficients, or problems for which the macro-scale PDE models may not be correct in localized regions and a micro-scale model must be used instead. Furthermore, we are interested in the long time solutions of such equations with waves traveling a long distance, such as the electromagnetic waves. The requirement on numerical methods is thus high order accuracy, low dissipation/dispersion errors over long time, and a clean and sharp discontinuity resolution. It would also be highly desirable to have the numerical methods to be flexible for complicated geometry and boundary conditions, and to be easy in an adaptive and massively parallel environment. The discontinuous Galerkin finite element method has this potential, however the design and analysis of the discontinuous Galerkin method for each specific class of partial differential equations would require detailed research. In this project we have addressed the following crucial algorithm issues: (1) design and analysis for stable and accurate discontinuous Galerkin methods for various linear and nonlinear partial differential equations from applications, including surface diffusion and Willmore flow of graphs, non-equilibrium flow equations, Boltzmann-Poisson systems in nano devices, Vlasov-Poisson system in plasma physics, compressible multi-medium flows, shallow water equations, equations with delta-singularities in flocking and other models, Schr\"{o}dinger equations, Helmholtz equations, Maxwell's equations in Drude metamaterials, Cahn-Hilliard equation, gradient flow problems with interaction potentials, kinetic models of self-organized dynamics, and incompressible flows; (2) study of robustness and accuracy issues including well balanced schemes, positivity-preserving and general bound-preserving techniques, and nonlinearly stable high order weighted essentially non-oscillatory (WENO) limiters to control spurious oscillations; (3) study of superconvergence of discontinuous Galerkin for linear and nonlinear partial differential equations, which provides important tools to guide adaptive computation to save computational costs; (4) study of efficient, stable and high order accurate time discretizations including implicit-explicit (IMEX) time discretizations, stable Lax-Wendroff type time discretizations. There are 66 refereed journal papers published, which have cited partial support from this DOE grant. The algorithms developed have found wide spread applications including applications by DOE lab scientists.},
doi = {10.2172/1499046},
journal = {},
number = ,
volume = ,
place = {United States},
year = {2019},
month = {3}
}

Works referenced in this record:

Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations
journal, June 2009


Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs
journal, December 2008


Fourier analysis for discontinuous Galerkin and related methods
journal, June 2009


High-order well-balanced schemes and applications to non-equilibrium flow
journal, October 2009


A discontinuous Galerkin solver for Boltzmann–Poisson systems in nano devices
journal, August 2009

  • Cheng, Yingda; Gamba, Irene M.; Majorana, Armando
  • Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 37-40
  • DOI: 10.1016/j.cma.2009.05.015

Analysis of a Local Discontinuous Galerkin Method for Linear Time-Dependent Fourth-Order Problems
journal, January 2009

  • Dong, Bo; Shu, Chi-Wang
  • SIAM Journal on Numerical Analysis, Vol. 47, Issue 5
  • DOI: 10.1137/080737472

Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation
journal, August 2011


Stability Analysis and A Priori Error Estimates of the Third Order Explicit Runge–Kutta Discontinuous Galerkin Method for Scalar Conservation Laws
journal, January 2010

  • Zhang, Qiang; Shu, Chi-Wang
  • SIAM Journal on Numerical Analysis, Vol. 48, Issue 3
  • DOI: 10.1137/090771363

An interface treating technique for compressible multi-medium flow with Runge–Kutta discontinuous Galerkin method
journal, November 2010


Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations
journal, December 2010


Central local discontinuous galerkin methods on overlapping cells for diffusion equations
journal, June 2011

  • Liu, Yingjie; Shu, Chi-Wang; Tadmor, Eitan
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 45, Issue 6
  • DOI: 10.1051/m2an/2011007

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations
journal, October 2011


High-order finite volume WENO schemes for the shallow water equations with dry states
journal, August 2011


Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations
journal, January 2011

  • Zhang, Yong-Tao; Chen, Shanqin; Li, Fengyan
  • SIAM Journal on Scientific Computing, Vol. 33, Issue 4
  • DOI: 10.1137/090770291

Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system
journal, November 2011

  • Shu, Chi-Wang; Carrillo, José; Ayuso, Blanca
  • Kinetic and Related Models, Vol. 4, Issue 4
  • DOI: 10.3934/krm.2011.4.955

Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension
journal, January 2012

  • Meng, X.; Shu, C. -W.; Wu, B.
  • IMA Journal of Numerical Analysis, Vol. 32, Issue 4
  • DOI: 10.1093/imanum/drr047

Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations
journal, January 2012

  • Xu, Yan; Shu, Chi-Wang
  • SIAM Journal on Numerical Analysis, Vol. 50, Issue 1
  • DOI: 10.1137/11082258X

Discontinuous Galerkin Methods for the Multi-Dimensional Vlasov–Poisson Problem
journal, October 2012

  • De Dios, Blanca Ayuso; Carrillo, JosÉ A.; Shu, Chi-Wang
  • Mathematical Models and Methods in Applied Sciences, Vol. 22, Issue 12
  • DOI: 10.1142/S021820251250042X

Superconvergence of Discontinuous Galerkin Methods for Scalar Nonlinear Conservation Laws in One Space Dimension
journal, January 2012

  • Meng, Xiong; Shu, Chi-Wang; Zhang, Qiang
  • SIAM Journal on Numerical Analysis, Vol. 50, Issue 5
  • DOI: 10.1137/110857635

Analysis of Optimal Superconvergence of Discontinuous Galerkin Method for Linear Hyperbolic Equations
journal, January 2012

  • Yang, Yang; Shu, Chi-Wang
  • SIAM Journal on Numerical Analysis, Vol. 50, Issue 6
  • DOI: 10.1137/110857647

A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods
journal, January 2013


Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes
journal, February 2013


Energy conserving local discontinuous Galerkin methods for wave propagation problems
journal, August 2013

  • Xing, Yulong; Chou, Ching-Shan; Shu, Chi-Wang
  • Inverse Problems and Imaging, Vol. 7, Issue 3
  • DOI: 10.3934/ipi.2013.7.967

Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes
journal, September 2013


Discontinuous Galerkin method for Krauseʼs consensus models and pressureless Euler equations
journal, November 2013


Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients
journal, January 2014


Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media
journal, September 2014


A Survey of High Order Schemes for the Shallow Water Equations
journal, June 2014


Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit-Explicit Time-Marching for Advection-Diffusion Problems
journal, January 2015

  • Wang, Haijin; Shu, Chi-Wang; Zhang, Qiang
  • SIAM Journal on Numerical Analysis, Vol. 53, Issue 1
  • DOI: 10.1137/140956750

Numerical Solution of the Viscous Surface Wave with Discontinuous Galerkin Method
journal, June 2015

  • Wu, Lei; Shu, Chi-Wang
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 49, Issue 4
  • DOI: 10.1051/m2an/2014065

Analysis of Sharp Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Parabolic Equations
journal, June 2015


Superconvergence of Discontinuous Galerkin Methods for Two-Dimensional Hyperbolic Equations
journal, January 2015

  • Cao, Waixiang; Shu, Chi-Wang; Yang, Yang
  • SIAM Journal on Numerical Analysis, Vol. 53, Issue 4
  • DOI: 10.1137/140996203

Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations
journal, September 2015

  • Meng, Xiong; Shu, Chi-Wang; Wu, Boying
  • Mathematics of Computation, Vol. 85, Issue 299
  • DOI: 10.1090/mcom/3022

A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation
journal, April 2015


Analysis of the local discontinuous Galerkin method for the drift-diffusion model of semiconductor devices
journal, July 2015


Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter
journal, April 2016


Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems
journal, July 2016

  • Wang, Haijin; Wang, Shiping; Zhang, Qiang
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, Issue 4
  • DOI: 10.1051/m2an/2015068

Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws
journal, May 2017

  • Sun, Zheng; Shu, Chi-Wang
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 3
  • DOI: 10.1051/m2an/2016049

Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws: Error Estimates To Smooth Solutions
journal, August 2016

  • Huang, Juntao; Shu, Chi-Wang
  • Numerical Methods for Partial Differential Equations, Vol. 33, Issue 2
  • DOI: 10.1002/num.22089

Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter on Unstructured Meshes
journal, February 2017


H(div) conforming and DG methods for incompressible Euler’s equations
journal, November 2016

  • Guzmán, Johnny; Shu, Chi-Wang; Sequeira, Filánder A.
  • IMA Journal of Numerical Analysis
  • DOI: 10.1093/imanum/drw054

A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model
journal, March 2017

  • Huang, Juntao; Shu, Chi-Wang
  • Mathematical Models and Methods in Applied Sciences, Vol. 27, Issue 03
  • DOI: 10.1142/S0218202517500099

A phase-based interior penalty discontinuous Galerkin method for the Helmholtz equation with spatially varying wavenumber
journal, May 2017

  • Lam, Chi Yeung; Shu, Chi-Wang
  • Computer Methods in Applied Mechanics and Engineering, Vol. 318
  • DOI: 10.1016/j.cma.2017.01.032

Optimal non-dissipative discontinuous Galerkin methods for Maxwell’s equations in Drude metamaterials
journal, April 2017


Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs
journal, September 2017

  • Wang, Haijin; Zhang, Qiang; Shu, Chi-Wang
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 5
  • DOI: 10.1051/m2an/2017017

Discontinuous Galerkin deterministic solvers for a Boltzmann–Poisson model of hot electron transport by averaged empirical pseudopotential band structures
journal, July 2017

  • Morales-Escalante, José; Gamba, Irene M.; Cheng, Yingda
  • Computer Methods in Applied Mechanics and Engineering, Vol. 321
  • DOI: 10.1016/j.cma.2017.03.003

Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients
journal, November 2017

  • Cao, Waixiang; Shu, Chi-Wang; Zhang, Zhimin
  • ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, Issue 6
  • DOI: 10.1051/m2an/2017026

A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials
journal, January 2018


Optimal Error Estimates of the Semidiscrete Central Discontinuous Galerkin Methods for Linear Hyperbolic Equations
journal, January 2018

  • Liu, Yong; Shu, Chi-Wang; Zhang, Mengping
  • SIAM Journal on Numerical Analysis, Vol. 56, Issue 1
  • DOI: 10.1137/16M1089484

Superconvergence of Discontinuous Galerkin Method for Scalar Nonlinear Hyperbolic Equations
journal, January 2018

  • Cao, Waixiang; Shu, Chi-Wang; Yang, Yang
  • SIAM Journal on Numerical Analysis, Vol. 56, Issue 2
  • DOI: 10.1137/17M1128605

Discontinuous Galerkin methods for a kinetic model of self-organized dynamics
journal, May 2018

  • Filbet, Francis; Shu, Chi-Wang
  • Mathematical Models and Methods in Applied Sciences, Vol. 28, Issue 06
  • DOI: 10.1142/S0218202518500318

Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow
journal, March 2018

  • Wang, Haijin; Liu, Yunxian; Zhang, Qiang
  • Mathematics of Computation, Vol. 88, Issue 315
  • DOI: 10.1090/mcom/3312

On the time growth of the error of the DG method for advective problems
journal, April 2018

  • Kučera, Václav; Shu, Chi-Wang
  • IMA Journal of Numerical Analysis, Vol. 39, Issue 2
  • DOI: 10.1093/imanum/dry013