Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity
Two finite difference discretization schemes for approximating the spatial derivatives in the diffusion equation in spherical coordinates with variable diffusivity are presented and analyzed. The numerical solutions obtained by the discretization schemes are compared for five cases of the functional...
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| Format: | Article |
| Language: | en_US |
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Elsevier
2017
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| Online Access: | http://hdl.handle.net/1721.1/106943 https://orcid.org/0000-0003-4304-3484 |
| _version_ | 1811080980924989440 |
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| author | Ford-Versypt, Ashlee Nicole Braatz, Richard D |
| author2 | Massachusetts Institute of Technology. Department of Chemical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Chemical Engineering Ford-Versypt, Ashlee Nicole Braatz, Richard D |
| author_sort | Ford-Versypt, Ashlee Nicole |
| collection | MIT |
| description | Two finite difference discretization schemes for approximating the spatial derivatives in the diffusion equation in spherical coordinates with variable diffusivity are presented and analyzed. The numerical solutions obtained by the discretization schemes are compared for five cases of the functional form for the variable diffusivity: (I) constant diffusivity, (II) temporally dependent diffusivity, (III) spatially dependent diffusivity, (IV) concentration-dependent diffusivity, and (V) implicitly defined, temporally and spatially dependent diffusivity. Although the schemes have similar agreement to known analytical or semi-analytical solutions in the first four cases, in the fifth case for the variable diffusivity, one scheme produces a stable, physically reasonable solution, while the other diverges. We recommend the adoption of the more accurate and stable of these finite difference discretization schemes to numerically approximate the spatial derivatives of the diffusion equation in spherical coordinates for any functional form of variable diffusivity, especially cases where the diffusivity is a function of position. |
| first_indexed | 2024-09-23T11:39:54Z |
| format | Article |
| id | mit-1721.1/106943 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:39:54Z |
| publishDate | 2017 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/1069432022-10-01T05:08:04Z Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity Ford-Versypt, Ashlee Nicole Braatz, Richard D Massachusetts Institute of Technology. Department of Chemical Engineering Ford-Versypt, Ashlee Nicole Braatz, Richard D Two finite difference discretization schemes for approximating the spatial derivatives in the diffusion equation in spherical coordinates with variable diffusivity are presented and analyzed. The numerical solutions obtained by the discretization schemes are compared for five cases of the functional form for the variable diffusivity: (I) constant diffusivity, (II) temporally dependent diffusivity, (III) spatially dependent diffusivity, (IV) concentration-dependent diffusivity, and (V) implicitly defined, temporally and spatially dependent diffusivity. Although the schemes have similar agreement to known analytical or semi-analytical solutions in the first four cases, in the fifth case for the variable diffusivity, one scheme produces a stable, physically reasonable solution, while the other diverges. We recommend the adoption of the more accurate and stable of these finite difference discretization schemes to numerically approximate the spatial derivatives of the diffusion equation in spherical coordinates for any functional form of variable diffusivity, especially cases where the diffusivity is a function of position. 2017-02-15T16:49:16Z 2017-02-15T16:49:16Z 2014-08 2014-04 Article http://purl.org/eprint/type/JournalArticle 0098-1354 http://hdl.handle.net/1721.1/106943 Ford Versypt, Ashlee N., and Richard D. Braatz. “Analysis of Finite Difference Discretization Schemes for Diffusion in Spheres with Variable Diffusivity.” Computers & Chemical Engineering 71 (2014): 241–252. https://orcid.org/0000-0003-4304-3484 en_US http://dx.doi.org/10.1016/j.compchemeng.2014.05.022 Computers & Chemical Engineering Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier PMC |
| spellingShingle | Ford-Versypt, Ashlee Nicole Braatz, Richard D Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| title | Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| title_full | Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| title_fullStr | Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| title_full_unstemmed | Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| title_short | Analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| title_sort | analysis of finite difference discretization schemes for diffusion in spheres with variable diffusivity |
| url | http://hdl.handle.net/1721.1/106943 https://orcid.org/0000-0003-4304-3484 |
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