A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation
The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration metho...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2015/539652 |
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| author | Wenyuan Liao |
| author_facet | Wenyuan Liao |
| author_sort | Wenyuan Liao |
| collection | DOAJ |
| description | The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method. |
| format | Article |
| id | doaj-art-afc61d8222e247f2aca3dbc202066662 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-afc61d8222e247f2aca3dbc2020666622025-02-03T01:06:56ZengWileyAbstract and Applied Analysis1085-33751687-04092015-01-01201510.1155/2015/539652539652A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion EquationWenyuan Liao0Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, CanadaThe semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.http://dx.doi.org/10.1155/2015/539652 |
| spellingShingle | Wenyuan Liao A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation Abstract and Applied Analysis |
| title | A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation |
| title_full | A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation |
| title_fullStr | A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation |
| title_full_unstemmed | A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation |
| title_short | A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation |
| title_sort | strongly a stable time integration method for solving the nonlinear reaction diffusion equation |
| url | http://dx.doi.org/10.1155/2015/539652 |
| work_keys_str_mv | AT wenyuanliao astronglyastabletimeintegrationmethodforsolvingthenonlinearreactiondiffusionequation AT wenyuanliao stronglyastabletimeintegrationmethodforsolvingthenonlinearreactiondiffusionequation |