Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein–Gordon–Schrödinger equations

Research output: Contribution to journalArticle

Abstract

The Klein–Gordon–Schrödinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank–Nicholson time discretization for the Schrödinger equation part, a Crank–Nicholson leap frog method in time for the Klein–Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein–Gordon or Schrödinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.

Original languageEnglish (US)
Pages (from-to)377-424
Number of pages48
JournalJournal of Applied Mathematics and Computing
Volume62
Issue number1-2
DOIs
StatePublished - Feb 1 2020

Fingerprint

Local Discontinuous Galerkin Method
Hamiltonians
Galerkin methods
Error Estimates
Energy
Conservation
Numerical methods
Scalar
Optimal Error Estimates
Klein-Gordon Equation
Time Discretization
Class
Mesons
Convergence Rate
Numerical Methods
Higher Order
Verify

Keywords

  • Energy conservation
  • Error estimates
  • Hamiltonian-preserving
  • Klein–Gordon–Schrödinger equations
  • Local discontinuous Galerkin methods

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

@article{3703d86937a4447fb8f89dbe4625ad92,
title = "Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein–Gordon–Schr{\"o}dinger equations",
abstract = "The Klein–Gordon–Schr{\"o}dinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank–Nicholson time discretization for the Schr{\"o}dinger equation part, a Crank–Nicholson leap frog method in time for the Klein–Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein–Gordon or Schr{\"o}dinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.",
keywords = "Energy conservation, Error estimates, Hamiltonian-preserving, Klein–Gordon–Schr{\"o}dinger equations, Local discontinuous Galerkin methods",
author = "He Yang",
year = "2020",
month = "2",
day = "1",
doi = "10.1007/s12190-019-01289-4",
language = "English (US)",
volume = "62",
pages = "377--424",
journal = "Journal of Applied Mathematics and Computing",
issn = "1598-5865",
publisher = "Springer Verlag",
number = "1-2",

}

TY - JOUR

T1 - Error estimates for a class of energy- and Hamiltonian-preserving local discontinuous Galerkin methods for the Klein–Gordon–Schrödinger equations

AU - Yang, He

PY - 2020/2/1

Y1 - 2020/2/1

N2 - The Klein–Gordon–Schrödinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank–Nicholson time discretization for the Schrödinger equation part, a Crank–Nicholson leap frog method in time for the Klein–Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein–Gordon or Schrödinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.

AB - The Klein–Gordon–Schrödinger (KGS) equations are classical models to describe the interaction between conservative scalar nucleons and neutral scalar mesons through Yukawa coupling. In this paper, we propose local discontinuous Galerkin (LDG) methods to solve the KGS equations. The methods involve a Crank–Nicholson time discretization for the Schrödinger equation part, a Crank–Nicholson leap frog method in time for the Klein–Gordon equation part, and local discontinuous Galerkin methods in space. Our designed numerical methods have high-order convergence rate, and energy- and Hamiltonian-preserving properties. We present the proofs of such conservation properties for both semi-discrete and fully-discrete schemes. We also establish optimal error estimates of the semi-discrete methods for the linearized KGS equations and the fully discrete methods for the KGS equations. The analysis can be extended to LDG methods for the nonlinear Klein–Gordon or Schrödinger equation, and the KGS equations in higher spatial dimensions. Several numerical tests are presented to verify some of our theoretical findings.

KW - Energy conservation

KW - Error estimates

KW - Hamiltonian-preserving

KW - Klein–Gordon–Schrödinger equations

KW - Local discontinuous Galerkin methods

UR - http://www.scopus.com/inward/record.url?scp=85073984997&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073984997&partnerID=8YFLogxK

U2 - 10.1007/s12190-019-01289-4

DO - 10.1007/s12190-019-01289-4

M3 - Article

AN - SCOPUS:85073984997

VL - 62

SP - 377

EP - 424

JO - Journal of Applied Mathematics and Computing

JF - Journal of Applied Mathematics and Computing

SN - 1598-5865

IS - 1-2

ER -