A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation
Two nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numeri...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/754543 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832556756863025152 |
|---|---|
| author | E. Momoniat |
| author_facet | E. Momoniat |
| author_sort | E. Momoniat |
| collection | DOAJ |
| description | Two nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numerical schemes are shown to preserve conserved quantities when compared to an implicit scheme in which the nonlinear term is approximated in the usual way. Comparisons to the single solitary wave solution show significantly better results, measured in the L2 and L∞ norms, when compared to results obtained using a Petrov-Galerkin finite element method and a splitted quadratic B-spline collocation method. The growth in the error when simulating the single solitary wave solution using the two “best” nonstandard numerical schemes is shown to be linear implying the nonstandard finite difference schemes are conservative. The formation of an undular bore for both steep and shallow initial profiles is captured without the formation of numerical instabilities. |
| format | Article |
| id | doaj-art-8dbe058090154dfbaac1ba0604a46176 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-8dbe058090154dfbaac1ba0604a461762025-02-03T05:44:28ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/754543754543A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave EquationE. Momoniat0Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South AfricaTwo nonstandard finite difference schemes are derived to solve the regularized long wave equation. The criteria for choosing the “best” nonstandard approximation to the nonlinear term in the regularized long wave equation come from considering the modified equation. The two “best” nonstandard numerical schemes are shown to preserve conserved quantities when compared to an implicit scheme in which the nonlinear term is approximated in the usual way. Comparisons to the single solitary wave solution show significantly better results, measured in the L2 and L∞ norms, when compared to results obtained using a Petrov-Galerkin finite element method and a splitted quadratic B-spline collocation method. The growth in the error when simulating the single solitary wave solution using the two “best” nonstandard numerical schemes is shown to be linear implying the nonstandard finite difference schemes are conservative. The formation of an undular bore for both steep and shallow initial profiles is captured without the formation of numerical instabilities.http://dx.doi.org/10.1155/2014/754543 |
| spellingShingle | E. Momoniat A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation Abstract and Applied Analysis |
| title | A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation |
| title_full | A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation |
| title_fullStr | A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation |
| title_full_unstemmed | A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation |
| title_short | A Modified Equation Approach to Selecting a Nonstandard Finite Difference Scheme Applied to the Regularized Long Wave Equation |
| title_sort | modified equation approach to selecting a nonstandard finite difference scheme applied to the regularized long wave equation |
| url | http://dx.doi.org/10.1155/2014/754543 |
| work_keys_str_mv | AT emomoniat amodifiedequationapproachtoselectinganonstandardfinitedifferenceschemeappliedtotheregularizedlongwaveequation AT emomoniat modifiedequationapproachtoselectinganonstandardfinitedifferenceschemeappliedtotheregularizedlongwaveequation |