Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering

The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modifie...

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Autores principales:	Romuald Szymkiewicz, Dariusz Gąsiorowski
Formato:	article
Lenguaje:	EN
Publicado:	IWA Publishing 2021
Materias:	1d and 2d advection–diffusion equation adaptive numerical scheme modified equation analysis numerical dissipation and dispersion strang splitting technique Information technology T58.5-58.64 Environmental technology. Sanitary engineering TD1-1066
Acceso en línea:	https://doaj.org/article/7edbbd541d084b9ca7cbadc0237fafd4
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spelling	oai:doaj.org-article:7edbbd541d084b9ca7cbadc0237fafd42021-11-23T18:48:45ZAdaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering1464-71411465-173410.2166/hydro.2021.062https://doaj.org/article/7edbbd541d084b9ca7cbadc0237fafd42021-11-01T00:00:00Zhttp://jh.iwaponline.com/content/23/6/1290https://doaj.org/toc/1464-7141https://doaj.org/toc/1465-1734The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations. HIGHLIGHTS Solution of 2D advection–diffusion equation using the Strang splitting method.; Solution of 1D advection–diffusion equation using the modified finite element method.; Modified equation analysis yields the coefficients of numerical dissipation and dispersion.; For variable flow velocities, the weighting parameter varies in space and time.; The adaptive algorithm ensuring the constant order of approximation up to the 4rd order.;Romuald SzymkiewiczDariusz GąsiorowskiIWA Publishingarticle1d and 2d advection–diffusion equationadaptive numerical schememodified equation analysisnumerical dissipation and dispersionstrang splitting techniqueInformation technologyT58.5-58.64Environmental technology. Sanitary engineeringTD1-1066ENJournal of Hydroinformatics, Vol 23, Iss 6, Pp 1290-1311 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	1d and 2d advection–diffusion equation adaptive numerical scheme modified equation analysis numerical dissipation and dispersion strang splitting technique Information technology T58.5-58.64 Environmental technology. Sanitary engineering TD1-1066
spellingShingle	1d and 2d advection–diffusion equation adaptive numerical scheme modified equation analysis numerical dissipation and dispersion strang splitting technique Information technology T58.5-58.64 Environmental technology. Sanitary engineering TD1-1066 Romuald Szymkiewicz Dariusz Gąsiorowski Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering
description	The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations. HIGHLIGHTS Solution of 2D advection–diffusion equation using the Strang splitting method.; Solution of 1D advection–diffusion equation using the modified finite element method.; Modified equation analysis yields the coefficients of numerical dissipation and dispersion.; For variable flow velocities, the weighting parameter varies in space and time.; The adaptive algorithm ensuring the constant order of approximation up to the 4rd order.;
format	article
author	Romuald Szymkiewicz Dariusz Gąsiorowski
author_facet	Romuald Szymkiewicz Dariusz Gąsiorowski
author_sort	Romuald Szymkiewicz
title	Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering
title_short	Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering
title_full	Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering
title_fullStr	Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering
title_full_unstemmed	Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering
title_sort	adaptive method for the solution of 1d and 2d advection–diffusion equations used in environmental engineering
publisher	IWA Publishing
publishDate	2021
url	https://doaj.org/article/7edbbd541d084b9ca7cbadc0237fafd4
work_keys_str_mv	AT romualdszymkiewicz adaptivemethodforthesolutionof1dand2dadvectiondiffusionequationsusedinenvironmentalengineering AT dariuszgasiorowski adaptivemethodforthesolutionof1dand2dadvectiondiffusionequationsusedinenvironmentalengineering
_version_	1718416161167638528

Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering

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