Development of adaptive methods for reaction-diffusion and other transport problems arising in electrochemistry

• Autorzy: Bieniasz L.

Warianty tytułu

PL

Rozwój metod adaptacyjnych dla zagadnień reakcji-dyfuzji i innych problemów transportu występujących w elektrochemii

• Języki publikacji: EN

Abstrakty

EN

Development of adaptive methods for reaction-diffusion and other transport problems arising in electrochemistry Lesław K. Bieniasz Institute of Physical Chemistry of the Polish Academy of Sciences, Department of Complex Systems and Chemical Processing of Information, ul. Niezapominajek 8, 30-239 Cracow, Poland. Tel. [phone], Fax. [phone], E-mail: [e-mail], URL: http://www.cyf-kr.edu.pl/~nbbienia, and Institute of Teleinformatics, Faculty of Electrical and Computer Engineering, Cracow University of Technology, ul. Warszawska 24, 31-155 Cracow, Poland. Computational modelling of reaction-diffusion and other reactive transport phenomena presents a challenging task in many areas of science and technology related to chemistry and biology, materials science not being excluded. Problems of this kind become particularly difficult to solve when the governing equations (for example partial or ordinary differential equations) are singularly perturbed, so that their solutions possess local layers or moving fronts. In such cases, adaptive methods that detect such difficult local solution structures, and appropriately concentrate the computational effort on resolving them, are necessary. Another reason for developing the adaptive methods is the modern trend towards automation of computational procedures: users of the simulation software want to obtain solutions having a guaranteed prescribed accuracy, independently of the location, extension and duration of the local solution structures, which may well not be known a priori. For the past 15 years, the present author has been developing finite-difference adaptive approaches to the numerical solution of reaction-diffusion and other reactive transport equations occurring in electrochemistry. The work concentrated on initial boundary value problems for partial differential equations in one-dimensional geometry [1-17], boundary value problems for ordinary differential equations [18-19], and recently on integral equations [20]. In the present communication the results of this work will be briefly summarized. Experiments with the patch-adaptive grid strategy [1-17] and with the local grid node insertion/deletion [18-19] will be used to demonstrate the advantages and disadvantages of the various methods. Some conclusions of potential interest to modellers in other areas will be attempted. In particular, it will be argued that much more work still has to be done to design satisfactory methods, even for spatially one-dimensional equations, despite the fact that for such problems the adaptive methodology is currently regarded to be mature.

PL

W ciągu ponad 15 lat pracy autora nad rozwojem adaptacyjnych metod różnic skończonych dla zagadnień reakcji-dyfuzji oraz innych zjawisk transportu reakcyjnego występujących w elektrochemii nagromadziło się wiele doświadczeń, które mogą być interesujące dla badaczy zajmujących się modelowaniem w innych dziedzinach, łącznie z nauką o materiałach. Wyniki tych badań zostaną krótko przedstawione, ze wskazaniem na wady i zalety różnych metod. Przedstawione zostaną argumenty na rzecz tezy, że potrzeba znacznie więcej pracy aby zaprojektować zadowalające metody adaptacyjne, nawet dla równań w przestrzeni jednowymiarowej, mimo iż obecnie uważa się, że metodologia adaptacyjna dla takich problemów osiągnęła stan dojrzały.

Słowa kluczowe

EN

computational electrochemistry; reaction-diffusion; finite-difference method; adaptive grids

• Wydawca: Wydawnictwa AGH
• Czasopismo: Computer Methods in Materials Science
• Rocznik: 2009
• Tom: Vol. 9, No. 2
• Strony: 296--301
• Opis fizyczny: Bibliogr. 15 poz., rys.

Twórcy

autor

Bieniasz L.

• Institute of Physical Chemistry of the Polish Academy os Sciences, Department of Complex Systems and Chemical Processing of Information, ul. Niezapominajek 8, 30-239 Kraków, Poland Institute of Teleinformatics, facultz of Electrical and Computer Enginee,

Bibliografia

• 1.      Bieniasz, L.K., Towards Computational Electrochemistry -a Kineticis's Perspective, in: Modern Aspects of Electrochemistry, vol. 35, eds. Conway, B. E., White, R. E., Kluwer/Plenum, New York. 2002, 135-195.
• 2.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations.   Part 1.   Introductory   Exploration   of  the   Finite-Difference Adaptive Moving Grid Solution of the One-Dimensional Fast Homogeneous Reaction-Difńision Problem with a Reaction Layer, J. Electroanal. Chem.,  360, 1993, 119-138.
• 3.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations.   Part 2.   An  Improved  Finite-Difference  Adaptive Moving Grid Technique for One-Dimensional Fast Homogeneous Reaction-Diffusion Problems with Reaction Layers at the Electrodes, J. Electroanal. Chem., 374, 1994, 1-22.
• 4.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations. Part 3. An Adaptive Moving Grid - Adaptive Time Step Strategy for Problems with Discontinuous Boundary Conditions at the Electrodes, J. Electroanal. Chem., 374, 1994,23-35.
• 5.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations. Part 4. The Adaptive Moving-Grid Solution of One-Dimensional Fast Homogeneous Reaction-Diffusion Problems with Extremely Thin Reaction Zones Away from the Electrodes, J. Electroanal. Chem., 379, 1994, 71-87.
• 6.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations. Part 5. A Finite-Difference, Adaptive Space/Time Grid Strategy Based on a Patch-Type Local Uniform Spatial   Grid   Refinement,   for   Kinetic   Models   in One Dimensional Space Geometry, J. Electroanal. Chem., 481, 2000, 115-133.
• 7.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations. Part 10. Extension of the Patch-Adaptive Strategy to Kinetic Models Involving Spatially Localised Unknowns at the Boundaries, Multiple Space Intervals, and Non-Local Boundary Conditions, in One-Dimensional Space Geometry, J. Electroanal. Chem., 527, 2002, 1-10.
• 8.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations. Part 14. Extension of the Patch-Adaptive Strategy to Time-Dependent   Models   Involving   Migration-Diffusion Transport in One-Dimensional Space Geometry, and Application to Example Transient Experiments Described by Nernst-Planck-Poisson Equations, J. Electroanal. Chem., 565,2004,251-271.
• 9.      Bieniasz, L.K., Use of Dynamically Adaptive Grid Techniques for the Solution of Electrochemical Kinetic Equations. Part 16: Patch-Adaptive Strategy Combined with the Extended  Numerov   Spatial  Discretisation,   Electrochim. Acta, 52, 2007, 3929-3940.
• 10.     Bieniasz, L.K., Experiments with a Local Adaptive Grid h-refinement for the Finite-Difference Solution of BVPs in Singularly Perturbed Second-Order ODEs, Appl. Math. Comput, 195,2008,196-219.
• 11.     Bieniasz, L.K., Adaptive Solution of BVPs in Singularly Perturbed Second-Order ODEs, by the Extended Numerov Method   Combined   with   an   Iterative   Local   Grid refinement, Appl. Math. Comput., 2008, 198, 665-682.
• 12.     Bieniasz, L.K., An Adaptive Huber Method with Local Error Control, for the Numerical Solution of the First Kind Abel Integral Equations, Computing, 83, 2008, 25-39.
• 13.     Bieniasz, L. K., Use of the Numerov Method to Improve the  Accuracy  of the   Spatial   Discretisation  in Finite -Difference Electrochemical Kinetic Simulations, Comput. Chem., 26, 2002, 633-644.
• 14.     Bieniasz, L. K., Improving the Accuracy of the Spatial Discretization in Finite-Difference Electrochemical Kinetic Simulations, by Means of the Extended Numerov Method, J. Comput. Chem., 25, 2004, 1075-1083.
• 15.    Carey, G. F., Computational Grids, Generation, Adaptation and Solution Strategies, Taylor and Francis, Washington, 1997.