Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature
The numerical integration of the time-dependent spherically-symmetric radial diffusion equation from a point source is considered. The flux through the source can vary in time, possibly stochastically based on the concentration produced by the source itself. Fick’s one-dimensional diffusion equation...
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| Format: | Online |
| Language: | English |
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Public Library of Science
2015
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| Online Access: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4514844/ |
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pubmed-4514844 |
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pubmed-45148442015-07-29 Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature Gillespie, Dirk Research Article The numerical integration of the time-dependent spherically-symmetric radial diffusion equation from a point source is considered. The flux through the source can vary in time, possibly stochastically based on the concentration produced by the source itself. Fick’s one-dimensional diffusion equation is integrated over a time interval by considering a source term and a propagation term. The source term adds new particles during the time interval, while the propagation term diffuses the concentration profile of the previous time step. The integral in the propagation term is evaluated numerically using a combination of a new diffusion-specific Gaussian quadrature and interpolation on a diffusion-specific grid. This attempts to balance accuracy with the least number of points for both integration and interpolation. The theory can also be extended to include a simple reaction-diffusion equation in the limit of high buffer concentrations. The method is unconditionally stable. In fact, not only does it converge for any time step Δt, the method offers one advantage over other methods because Δt can be arbitrarily large; it is solely defined by the timescale on which the flux source turns on and off. Public Library of Science 2015-07-24 /pmc/articles/PMC4514844/ /pubmed/26208111 http://dx.doi.org/10.1371/journal.pone.0132273 Text en © 2015 Dirk Gillespie http://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited. |
| repository_type |
Open Access Journal |
| institution_category |
Foreign Institution |
| institution |
US National Center for Biotechnology Information |
| building |
NCBI PubMed |
| collection |
Online Access |
| language |
English |
| format |
Online |
| author |
Gillespie, Dirk |
| spellingShingle |
Gillespie, Dirk Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature |
| author_facet |
Gillespie, Dirk |
| author_sort |
Gillespie, Dirk |
| title |
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature |
| title_short |
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature |
| title_full |
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature |
| title_fullStr |
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature |
| title_full_unstemmed |
Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature |
| title_sort |
algorithm for the time-propagation of the radial diffusion equation based on a gaussian quadrature |
| description |
The numerical integration of the time-dependent spherically-symmetric radial diffusion equation from a point source is considered. The flux through the source can vary in time, possibly stochastically based on the concentration produced by the source itself. Fick’s one-dimensional diffusion equation is integrated over a time interval by considering a source term and a propagation term. The source term adds new particles during the time interval, while the propagation term diffuses the concentration profile of the previous time step. The integral in the propagation term is evaluated numerically using a combination of a new diffusion-specific Gaussian quadrature and interpolation on a diffusion-specific grid. This attempts to balance accuracy with the least number of points for both integration and interpolation. The theory can also be extended to include a simple reaction-diffusion equation in the limit of high buffer concentrations. The method is unconditionally stable. In fact, not only does it converge for any time step Δt, the method offers one advantage over other methods because Δt can be arbitrarily large; it is solely defined by the timescale on which the flux source turns on and off. |
| publisher |
Public Library of Science |
| publishDate |
2015 |
| url |
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4514844/ |
| _version_ |
1613252094032084992 |