An Eulerian-Lagrangian control volume scheme for two-dimensional unsteady advection-diffusion problems

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Abstract

We develop a mass conservative Eulerian-Lagrangian control volume scheme (ELCVS) for the solution of the transient advection-diffusion equations in two space dimensions. This method uses finite volume test functions over the space-time domain defined by the characteristics within the framework of the class of Eulerian-Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight-node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well-received classes of numerical methods for the equations considered.

Original languageEnglish
Pages (from-to)1481-1496
Number of pages16
JournalNumerical Methods for Partial Differential Equations
Volume28
Issue number5
DOIs
Publication statusPublished - Sep 2012

Fingerprint

Advection-diffusion
Control Volume
Diffusion Problem
Advection
Conservation
Adjoint Method
Characteristics Method
Program processors
Numerical methods
Finite Volume
Boundary conditions
Mass Conservation
Grid
Advection-diffusion Equation
Method of Characteristics
Test function
Simplify
Space-time
Numerical Methods
Class

Keywords

  • advection diffusion equations
  • characteristic methods
  • control volume methods
  • Eulerian-Lagrangian methods

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics
  • Analysis

Cite this

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abstract = "We develop a mass conservative Eulerian-Lagrangian control volume scheme (ELCVS) for the solution of the transient advection-diffusion equations in two space dimensions. This method uses finite volume test functions over the space-time domain defined by the characteristics within the framework of the class of Eulerian-Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight-node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well-received classes of numerical methods for the equations considered.",
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AB - We develop a mass conservative Eulerian-Lagrangian control volume scheme (ELCVS) for the solution of the transient advection-diffusion equations in two space dimensions. This method uses finite volume test functions over the space-time domain defined by the characteristics within the framework of the class of Eulerian-Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight-node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well-received classes of numerical methods for the equations considered.

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