Services on Demand
Journal
Article
Indicators
- Cited by SciELO
- Access statistics
Related links
- Similars in SciELO
Share
Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.22 n.1 San José Jan./Jun. 2015
An adaptive wavelet-galerkin method for parabolic partial differential equations
Un método wavelet-galerkin adaptativo para ecuaciones diferenciales parciales parabólicas
Un método wavelet-galerkin adaptativo para ecuaciones diferenciales parciales parabólicas
Abstract
In this paper an Adaptive Wavelet-Galerkin method for the solution of parabolic partial differential equations modeling physical problems with different spatial and temporal scales is developed. A semi-implicit time difference scheme is applied and B-spline multiresolution structure on the interval is used. As in many cases these solutions are known to present localized sharp gradients, local error estimators are designed and an efficient adaptive strategy to choose the appropriate scale for each time is developed. Finally, experiments were performed to illustrate the applicability and efficiency of the proposed method.
Keywords: B-spline; multiresolution analysis; wavelet-Galerkin.
Resumen
En este trabajo se desarrolla un método Wavelet-Galerkin Adaptativo para la resolución de ecuaciones diferenciales parabólicas que modelan problemas físicos, con diferentes escalas en el espacio y en el tiempo. Se utiliza un esquema semi-implícito en diferencias temporales y la estructura multirresolución de las B-splines sobre intervalo. Como es sabido que en muchos casos las soluciones presentan gradientes localmente altos, se han diseñado estimadores locales de error y una estrategia adaptativa eficiente para elegir la escala apropiada en cada tiempo. Finalmente, se realizaron experimentos que ilustran la aplicabilidad y la eficiencia del método propuesto.
Palabras clave: B-spline, análisis multirresolución; wavelet-Galerkin; ondeletas Galerkin.
Mathematics Subject Classification: 65M99.
Ver contenido en pdf.
References
[1] Bertoluzza, S.; Naldi, G. (1996) “A wavelet collocation method for the numerical solution of partial differential equations”, Applied and Computational Harmonic Analysis 3(1): 1–9. [ Links ]
[2] Bindal, A.; Khinast, J.G.; Ierapetritou, M.G. (2003) “Adaptive multiscale solution of dynamical systems in chemical processes using wavelets”, Computers and Chemical Engineering 27(1): 131–142. [ Links ]
[3] Burgers, J.M. (1948) “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech. 1: 171–199. [ Links ]
[4] Cammilleri, A; Serrano, E.P. (2001) “Spline multiresolution analysis on the interval”, Latin American Applied Research 31(2): 65–71. [ Links ]
[5] Chui, C.K. (1992) An Introduction toWavelets. Academic Press, New York. [ Links ]
[6] Ciarlet, P.G. (1978) The Finite Element Method for Elliptic Problems. North Holland, New York. [ Links ]
[7] Kumar, B V.; Mehra, M. (2005) “Wavelet-Taylor Galerkin method for the Burgers equation”, BIT Numerical Mathematics 45: 543–560. [ Links ]
8 Kumar, V.; Mehra, M. (2007) "Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction diffusion problems", International Journal of Wavelets, Multiresolution and Information Processing 5: 317–331. [ Links ]
[9] Lin, E.B.; Zhou, X. (2001) “Connection coefficients on an interval and wavelet solutions of Burgers equation”, Journal of Computational and Applied Mathematics 135(1): 63–78. [ Links ]
[10] Mallat, S.G. (2009) A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press - Elsevier, MA EE.UU. [ Links ]
[11] Quraishi, S.M.; Gupta, R.; Sandeep, K. (2009) “Adaptive wavelet Galerkin solution of some elastostatics problems on irregularly spaced nodes”, The Open Numerical Methods Journal 1: 20–26. [ Links ]
[12] Schöenberg, I.J. (1969) “Cardinal interpolation and spline functions”, Journal of Approximation Theory 2: 167–206. [ Links ]
[13] Schult, T.L.;Wyld, H.W. (1992) “Using wavelets to solve the Burgers equation: A comparative study”, Physical Review A 46(12): 7953–7958. [ Links ]
[14] Vampa, V.; Martín, M.T.; Serrano, E. (2010) “A hybrid method using wavelets for the numerical solution of boundary value problems on the interval”, Appl. Math. Comput. 217(7): 3355–3367. [ Links ]
[15] Vampa, V. (2011) Desarrollo de Herramientas Basadas en la Transformada Wavelet para su Aplicación en la Resolución Numérica de Ecuaciones Diferenciales. Tesis de Doctorado en Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina. [ Links ]
[16] Vampa, V.; Martín, M.T.; Serrano, E. (2012) “A new refinement Wavelet-Galerkin method in a spline local multiresolution analysis scheme for boundary value problems”, Int. Journal of Wavelets, Multiresolution and Information Processing 11(2), 19 pp. [ Links ]
17 Vasilyev, O.V.; Paolucci, S. (1996) “A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain”, Journal of Computational Physics 125(2): 498–512. [ Links ]
[18] Walnut, D.F. (2002) An Introduction to Wavelet Analysis. Applied and Numerical Harmonic Analysis Series, Birkhäuser, Boston. [ Links ]
[19] Whitham, G.B. (1974) Linear and Nonlinear Waves, Wiley, New York. [ Links ]
[2] Bindal, A.; Khinast, J.G.; Ierapetritou, M.G. (2003) “Adaptive multiscale solution of dynamical systems in chemical processes using wavelets”, Computers and Chemical Engineering 27(1): 131–142. [ Links ]
[3] Burgers, J.M. (1948) “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech. 1: 171–199. [ Links ]
[4] Cammilleri, A; Serrano, E.P. (2001) “Spline multiresolution analysis on the interval”, Latin American Applied Research 31(2): 65–71. [ Links ]
[5] Chui, C.K. (1992) An Introduction toWavelets. Academic Press, New York. [ Links ]
[6] Ciarlet, P.G. (1978) The Finite Element Method for Elliptic Problems. North Holland, New York. [ Links ]
[7] Kumar, B V.; Mehra, M. (2005) “Wavelet-Taylor Galerkin method for the Burgers equation”, BIT Numerical Mathematics 45: 543–560. [ Links ]
8 Kumar, V.; Mehra, M. (2007) "Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction diffusion problems", International Journal of Wavelets, Multiresolution and Information Processing 5: 317–331. [ Links ]
[9] Lin, E.B.; Zhou, X. (2001) “Connection coefficients on an interval and wavelet solutions of Burgers equation”, Journal of Computational and Applied Mathematics 135(1): 63–78. [ Links ]
[10] Mallat, S.G. (2009) A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press - Elsevier, MA EE.UU. [ Links ]
[11] Quraishi, S.M.; Gupta, R.; Sandeep, K. (2009) “Adaptive wavelet Galerkin solution of some elastostatics problems on irregularly spaced nodes”, The Open Numerical Methods Journal 1: 20–26. [ Links ]
[12] Schöenberg, I.J. (1969) “Cardinal interpolation and spline functions”, Journal of Approximation Theory 2: 167–206. [ Links ]
[13] Schult, T.L.;Wyld, H.W. (1992) “Using wavelets to solve the Burgers equation: A comparative study”, Physical Review A 46(12): 7953–7958. [ Links ]
[14] Vampa, V.; Martín, M.T.; Serrano, E. (2010) “A hybrid method using wavelets for the numerical solution of boundary value problems on the interval”, Appl. Math. Comput. 217(7): 3355–3367. [ Links ]
[15] Vampa, V. (2011) Desarrollo de Herramientas Basadas en la Transformada Wavelet para su Aplicación en la Resolución Numérica de Ecuaciones Diferenciales. Tesis de Doctorado en Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina. [ Links ]
[16] Vampa, V.; Martín, M.T.; Serrano, E. (2012) “A new refinement Wavelet-Galerkin method in a spline local multiresolution analysis scheme for boundary value problems”, Int. Journal of Wavelets, Multiresolution and Information Processing 11(2), 19 pp. [ Links ]
17 Vasilyev, O.V.; Paolucci, S. (1996) “A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain”, Journal of Computational Physics 125(2): 498–512. [ Links ]
[18] Walnut, D.F. (2002) An Introduction to Wavelet Analysis. Applied and Numerical Harmonic Analysis Series, Birkhäuser, Boston. [ Links ]
[19] Whitham, G.B. (1974) Linear and Nonlinear Waves, Wiley, New York. [ Links ]
*Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata, Argentina. E-Mail: victoriavampa@gmail.com
†Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina. IFLP-CCTCONICET, C. C. 727, 1900 La Plata, Argentina. E-Mail: teremartin.map@gmail.com
Received: 25/Feb/2014; Revised: 28/Aug/2014; Accepted: 19/Sep/2014