Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations
It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when t...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Western Libraries
2024-12-01
|
| Series: | Mathematics in Applied Sciences and Engineering |
| Subjects: | |
| Online Access: | https://ojs.lib.uwo.ca/index.php/mase/article/view/20176 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1841560713110224896 |
|---|---|
| author | Chinedu Nwaigwe |
| author_facet | Chinedu Nwaigwe |
| author_sort | Chinedu Nwaigwe |
| collection | DOAJ |
| description |
It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when they can, or require larger stencil to maintain high-order accuracy. This is challenging, especially due to the convection term which can easily lead to oscillatory solutions if naively approximated. This paper proposes a spatially second-order scheme which is able to simulate purely-convection problems with high-order accuracy on minimal stencil. Our idea is based on non-standard central discretization of the convection term. We first discretize the diffusion term in both space and time but discretize the convection term in space only. Next, the semi-discrete convection term is split into positive and negative parts. Transport coefficients are evaluated explicitly in time while different spatial operators are discretized either implicitly or explicitly such that positivity of canonical form is guaranteed. This led to a second-order scheme with all the above desirable properties. Under smoothness assumptions consistency is proved, discrete maximum principle is used to derived two-sided bounds on the numerical solution, and convergence is proved in maximum norm. Several numerical examples are provided to verify the second-order spatial accuracy, convergence and the ability to simulate purely convection problems.
|
| format | Article |
| id | doaj-art-890366980aab4b229c91a6343901cd6e |
| institution | Kabale University |
| issn | 2563-1926 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Western Libraries |
| record_format | Article |
| series | Mathematics in Applied Sciences and Engineering |
| spelling | doaj-art-890366980aab4b229c91a6343901cd6e2025-01-03T18:35:34ZengWestern LibrariesMathematics in Applied Sciences and Engineering2563-19262024-12-015410.5206/mase/20176Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equationsChinedu Nwaigwe0Rivers State University It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when they can, or require larger stencil to maintain high-order accuracy. This is challenging, especially due to the convection term which can easily lead to oscillatory solutions if naively approximated. This paper proposes a spatially second-order scheme which is able to simulate purely-convection problems with high-order accuracy on minimal stencil. Our idea is based on non-standard central discretization of the convection term. We first discretize the diffusion term in both space and time but discretize the convection term in space only. Next, the semi-discrete convection term is split into positive and negative parts. Transport coefficients are evaluated explicitly in time while different spatial operators are discretized either implicitly or explicitly such that positivity of canonical form is guaranteed. This led to a second-order scheme with all the above desirable properties. Under smoothness assumptions consistency is proved, discrete maximum principle is used to derived two-sided bounds on the numerical solution, and convergence is proved in maximum norm. Several numerical examples are provided to verify the second-order spatial accuracy, convergence and the ability to simulate purely convection problems. https://ojs.lib.uwo.ca/index.php/mase/article/view/20176Convection-diffusion equations, nonlinear coefficients, two-sided bounds, experimental order of convergence, consistency, freezing coefficients. |
| spellingShingle | Chinedu Nwaigwe Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations Mathematics in Applied Sciences and Engineering Convection-diffusion equations, nonlinear coefficients, two-sided bounds, experimental order of convergence, consistency, freezing coefficients. |
| title | Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations |
| title_full | Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations |
| title_fullStr | Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations |
| title_full_unstemmed | Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations |
| title_short | Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations |
| title_sort | compact scheme with two sided estimates for nonlinear convection diffusion reaction equations |
| topic | Convection-diffusion equations, nonlinear coefficients, two-sided bounds, experimental order of convergence, consistency, freezing coefficients. |
| url | https://ojs.lib.uwo.ca/index.php/mase/article/view/20176 |
| work_keys_str_mv | AT chinedunwaigwe compactschemewithtwosidedestimatesfornonlinearconvectiondiffusionreactionequations |