Remarks on the numerical approximation of Dirac delta functions

We investigate the convergence rate of the solutions of one and two-dimensional Poisson-type PDEs where the Dirac delta function, representing the forcing term, is approximated by several expressions.The goal is to see if the solution to a Poisson’s equation converges when solved by a numerical meth...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	Vincenzo Schiano Di Cola, Salvatore Cuomo, Gerardo Severino
Formato:	article
Lenguaje:	EN
Publicado:	Elsevier 2021
Materias:	Approximate Dirac delta function Immersed boundary (IB) method Level set methods Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/8f9fdfb8d82e49e684f25e3a5d3c4cab
Etiquetas:	Agregar Etiqueta Sin Etiquetas, Sea el primero en etiquetar este registro!

id	oai:doaj.org-article:8f9fdfb8d82e49e684f25e3a5d3c4cab
record_format	dspace
spelling	oai:doaj.org-article:8f9fdfb8d82e49e684f25e3a5d3c4cab2021-11-14T04:35:18ZRemarks on the numerical approximation of Dirac delta functions2590-037410.1016/j.rinam.2021.100200https://doaj.org/article/8f9fdfb8d82e49e684f25e3a5d3c4cab2021-11-01T00:00:00Zhttp://www.sciencedirect.com/science/article/pii/S259003742100042Xhttps://doaj.org/toc/2590-0374We investigate the convergence rate of the solutions of one and two-dimensional Poisson-type PDEs where the Dirac delta function, representing the forcing term, is approximated by several expressions.The goal is to see if the solution to a Poisson’s equation converges when solved by a numerical method, with a source or sink approximated using a delta proposed in the literature. We will look at two parameters, how fast it converges, by estimating the order of convergence, and how well, by calculating the error between the analytical form and the numerical result. We investigate smoothed discrete delta functions based on the Immersed boundary (IB) approach, and we revisit their definitions, as in level set methods, by expanding their support for assessing higher-order of convergence in PDE solutions. We developed a Python package utilizing FiPy, a PDE solver based on the finite volume (FV) technique, and accelerated the solver with the AmgX package, a GPU solution. We have observed that when the support is wider, better results may be achieved. Moreover, the overall trends of error and the convergence rate in the 2D configuration differ from the 1D problems.Vincenzo Schiano Di ColaSalvatore CuomoGerardo SeverinoElsevierarticleApproximate Dirac delta functionImmersed boundary (IB) methodLevel set methodsMathematicsQA1-939ENResults in Applied Mathematics, Vol 12, Iss , Pp 100200- (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Approximate Dirac delta function Immersed boundary (IB) method Level set methods Mathematics QA1-939
spellingShingle	Approximate Dirac delta function Immersed boundary (IB) method Level set methods Mathematics QA1-939 Vincenzo Schiano Di Cola Salvatore Cuomo Gerardo Severino Remarks on the numerical approximation of Dirac delta functions
description	We investigate the convergence rate of the solutions of one and two-dimensional Poisson-type PDEs where the Dirac delta function, representing the forcing term, is approximated by several expressions.The goal is to see if the solution to a Poisson’s equation converges when solved by a numerical method, with a source or sink approximated using a delta proposed in the literature. We will look at two parameters, how fast it converges, by estimating the order of convergence, and how well, by calculating the error between the analytical form and the numerical result. We investigate smoothed discrete delta functions based on the Immersed boundary (IB) approach, and we revisit their definitions, as in level set methods, by expanding their support for assessing higher-order of convergence in PDE solutions. We developed a Python package utilizing FiPy, a PDE solver based on the finite volume (FV) technique, and accelerated the solver with the AmgX package, a GPU solution. We have observed that when the support is wider, better results may be achieved. Moreover, the overall trends of error and the convergence rate in the 2D configuration differ from the 1D problems.
format	article
author	Vincenzo Schiano Di Cola Salvatore Cuomo Gerardo Severino
author_facet	Vincenzo Schiano Di Cola Salvatore Cuomo Gerardo Severino
author_sort	Vincenzo Schiano Di Cola
title	Remarks on the numerical approximation of Dirac delta functions
title_short	Remarks on the numerical approximation of Dirac delta functions
title_full	Remarks on the numerical approximation of Dirac delta functions
title_fullStr	Remarks on the numerical approximation of Dirac delta functions
title_full_unstemmed	Remarks on the numerical approximation of Dirac delta functions
title_sort	remarks on the numerical approximation of dirac delta functions
publisher	Elsevier
publishDate	2021
url	https://doaj.org/article/8f9fdfb8d82e49e684f25e3a5d3c4cab
work_keys_str_mv	AT vincenzoschianodicola remarksonthenumericalapproximationofdiracdeltafunctions AT salvatorecuomo remarksonthenumericalapproximationofdiracdeltafunctions AT gerardoseverino remarksonthenumericalapproximationofdiracdeltafunctions
_version_	1718429909907406848

Remarks on the numerical approximation of Dirac delta functions

Ejemplares similares