Spatially-dependent and nonlinear fluid transport: coupling framework

Jürgen Geiser

Open Mathematics (2012)

  • Volume: 10, Issue: 1, page 116-136
  • ISSN: 2391-5455

Abstract

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We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas. The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation methods or nonlinear solver methods. The nonlinear methods are based on characteristic methods and can be generalized numerically to higher-order TVD methods [Harten A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 1983, 49(3), 357–393]. In this article we will focus on the derivation of some analytical solutions for spatially dependent and nonlinear problems which can be embedded into finite volume methods. The main contribution is to embed one-dimensional analytical solutions into multi-dimensional finite volume methods with the construction idea of mass transport [Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922]. At the end of the article we present some results of numerical experiments for different benchmark problems.

How to cite

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Jürgen Geiser. "Spatially-dependent and nonlinear fluid transport: coupling framework." Open Mathematics 10.1 (2012): 116-136. <http://eudml.org/doc/269164>.

@article{JürgenGeiser2012,
abstract = {We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas. The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation methods or nonlinear solver methods. The nonlinear methods are based on characteristic methods and can be generalized numerically to higher-order TVD methods [Harten A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 1983, 49(3), 357–393]. In this article we will focus on the derivation of some analytical solutions for spatially dependent and nonlinear problems which can be embedded into finite volume methods. The main contribution is to embed one-dimensional analytical solutions into multi-dimensional finite volume methods with the construction idea of mass transport [Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922]. At the end of the article we present some results of numerical experiments for different benchmark problems.},
author = {Jürgen Geiser},
journal = {Open Mathematics},
keywords = {Advection-reaction equation; Spatial and nonlinear transport; Laplace transformation; Analytical solutions; Finite volume methods; advection-reaction equation; spatial and nonlinear transport; analytical solutions; finite volume methods},
language = {eng},
number = {1},
pages = {116-136},
title = {Spatially-dependent and nonlinear fluid transport: coupling framework},
url = {http://eudml.org/doc/269164},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Jürgen Geiser
TI - Spatially-dependent and nonlinear fluid transport: coupling framework
JO - Open Mathematics
PY - 2012
VL - 10
IS - 1
SP - 116
EP - 136
AB - We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas. The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation methods or nonlinear solver methods. The nonlinear methods are based on characteristic methods and can be generalized numerically to higher-order TVD methods [Harten A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 1983, 49(3), 357–393]. In this article we will focus on the derivation of some analytical solutions for spatially dependent and nonlinear problems which can be embedded into finite volume methods. The main contribution is to embed one-dimensional analytical solutions into multi-dimensional finite volume methods with the construction idea of mass transport [Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922]. At the end of the article we present some results of numerical experiments for different benchmark problems.
LA - eng
KW - Advection-reaction equation; Spatial and nonlinear transport; Laplace transformation; Analytical solutions; Finite volume methods; advection-reaction equation; spatial and nonlinear transport; analytical solutions; finite volume methods
UR - http://eudml.org/doc/269164
ER -

References

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  7. [7] Geiser J., Discretisation Methods for Systems of Convective-Diffusive Dispersive-Reactive Equations and Applications, PhD thesis, Universität Heidelberg, 2004 
  8. [8] Geiser J., Discretization methods with embedded analytical solutions for convection-diffusion dispersion-reaction equations and applications, J. Engrg. Math., 2007, 57(1), 79–98 http://dx.doi.org/10.1007/s10665-006-9057-y Zbl1148.65067
  9. [9] Geiser J., Mobile and immobile fluid transport: coupling framework, Internat. J. Numer. Methods Fluids, 2010, 65(8), 877–922 http://dx.doi.org/10.1002/fld.2225 Zbl05862336
  10. [10] Geiser J., Zacher T., Time dependent fluid transport: analytical framework. preprint available at http://webdoc.sub.gwdg.de/ebook/serien/e/preprint_HUB/P-11-05.pdf 
  11. [11] Higashi K., Pigford T.H., Analytical models for migration of radionuclides in geologic sorbing media, Journal of Nuclear Science and Technology, 1980, 17(9), 700–709 http://dx.doi.org/10.3327/jnst.17.700 
  12. [12] Kelley C.T., Iterative Methods for Linear and Nonlinear Equations, Frontiers Appl. Math., 16, SIAM, Philadelphia, 1995 http://dx.doi.org/10.1137/1.9781611970944 Zbl0832.65046
  13. [13] LeVeque R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2002 http://dx.doi.org/10.1017/CBO9780511791253 Zbl1010.65040
  14. [14] Van Genuchten M.T, Convective-dispersive transport of solutes involved in sequential first-order decay reactions, Computers & Geosciences, 1985, 11(2), 129–147 http://dx.doi.org/10.1016/0098-3004(85)90003-2 

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