Runge-Kutta Characteristic Methods for First-Order Linear Hyperbolic Equations

Hong Wang, Mohamed Al-Lawatia, Aleksey S. Telyakovskiy

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We develop two Runge-Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian-Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov-Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge-Kutta methods developed in this article.

Original languageEnglish
Pages (from-to)617-661
Number of pages45
JournalNumerical Methods for Partial Differential Equations
Volume13
Issue number6
Publication statusPublished - Nov 1997

Fingerprint

Linear Hyperbolic Equation
Characteristics Method
Galerkin methods
Runge-Kutta Methods
Boundary conditions
First-order
Runge Kutta methods
Boundary value problems
Streamline Diffusion
Artificial Boundary Conditions
Adjoint Method
Petrov-Galerkin Method
Backtracking
Discontinuous Galerkin Method
Time Integration
Positive definite
Experiments
Initial-boundary-value Problem
Rate of Convergence
Numerical Experiment

Keywords

  • Characteristic methods
  • Eulerian-Lagrangian methods
  • Numerical solution of first-order hyperbolic equations
  • Rung-Kutta methods

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Computational Mathematics

Cite this

Runge-Kutta Characteristic Methods for First-Order Linear Hyperbolic Equations. / Wang, Hong; Al-Lawatia, Mohamed; Telyakovskiy, Aleksey S.

In: Numerical Methods for Partial Differential Equations, Vol. 13, No. 6, 11.1997, p. 617-661.

Research output: Contribution to journalArticle

@article{afa984ef7c044a5597b81e5af6f53cef,
title = "Runge-Kutta Characteristic Methods for First-Order Linear Hyperbolic Equations",
abstract = "We develop two Runge-Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian-Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov-Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge-Kutta methods developed in this article.",
keywords = "Characteristic methods, Eulerian-Lagrangian methods, Numerical solution of first-order hyperbolic equations, Rung-Kutta methods",
author = "Hong Wang and Mohamed Al-Lawatia and Telyakovskiy, {Aleksey S.}",
year = "1997",
month = "11",
language = "English",
volume = "13",
pages = "617--661",
journal = "Numerical Methods for Partial Differential Equations",
issn = "0749-159X",
publisher = "John Wiley and Sons Inc.",
number = "6",

}

TY - JOUR

T1 - Runge-Kutta Characteristic Methods for First-Order Linear Hyperbolic Equations

AU - Wang, Hong

AU - Al-Lawatia, Mohamed

AU - Telyakovskiy, Aleksey S.

PY - 1997/11

Y1 - 1997/11

N2 - We develop two Runge-Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian-Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov-Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge-Kutta methods developed in this article.

AB - We develop two Runge-Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian-Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov-Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge-Kutta methods developed in this article.

KW - Characteristic methods

KW - Eulerian-Lagrangian methods

KW - Numerical solution of first-order hyperbolic equations

KW - Rung-Kutta methods

UR - http://www.scopus.com/inward/record.url?scp=0001417548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001417548&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0001417548

VL - 13

SP - 617

EP - 661

JO - Numerical Methods for Partial Differential Equations

JF - Numerical Methods for Partial Differential Equations

SN - 0749-159X

IS - 6

ER -