A locally conservative Eulerian-Lagrangian control-volume method for transient advection-diffusion equations

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11 Citations (Scopus)

Abstract

Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection-diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian-Lagrangian control-volume method (ELCVM) for transient advection-diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well-regarded Eulerian-Lagrangian methods, which were previously shown to be very competitive with many well-perceived methods.

Original languageEnglish
Pages (from-to)577-599
Number of pages23
JournalNumerical Methods for Partial Differential Equations
Volume22
Issue number3
DOIs
Publication statusPublished - May 2006

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Advection-diffusion Equation
Control Volume
Advection
Method of Characteristics
Polynomials
Numerical Solution
Eulerian-Lagrangian Methods
Curved Boundary
Characteristics Method
Piecewise Polynomials
Experiments
Numerical Experiment
Grid
Alternatives
Evaluation

Keywords

  • Advection-diffusion equations
  • Alternating-direction splitting
  • Finite-volume method
  • Maximum principle
  • Upwind method

ASJC Scopus subject areas

  • Applied Mathematics
  • Analysis
  • Computational Mathematics

Cite this

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N2 - Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection-diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian-Lagrangian control-volume method (ELCVM) for transient advection-diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well-regarded Eulerian-Lagrangian methods, which were previously shown to be very competitive with many well-perceived methods.

AB - Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection-diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian-Lagrangian control-volume method (ELCVM) for transient advection-diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well-regarded Eulerian-Lagrangian methods, which were previously shown to be very competitive with many well-perceived methods.

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KW - Maximum principle

KW - Upwind method

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