Resumen de tesis doctoral

Accurate Flexible Numerical Boundary Conditions for Multidimensional Transport and Diffusion

Precisas Flexibles Condiciones de Frontera Numéricas para el Transporte y Difusión Multidimensional

Graduated: Dr. Denis Filatov
Centre for Computing Research (CIC),
National Polytechnic Institute (IPN),
Av. Juan de Dios Batiz s/n,
C.P. 07738, Mexico, D.F.
E–mail: denisfilatov@mail.ru

Advisor: Prof. Dr. Mikhail Alexandrov
Centre for Computing Research (CIC),
National Polytechnic Institute (IPN),
Av. Juan de Dios Batiz s/n,
C.P. 07738, Mexico, D.F.
E–mail: dynerl950@mail.ru

Co–Advisor: Prof. Dr. Sc. Mikhail Yudin
Moscow State Geological Prospecting University (MGRI–MGGU),
23 Miklukho–Maklaya st, 117997, Moscow, Russia
E–mail: judin@msgpa.ru

Graduated on: January 20, 2004

Abstract

A method for numerical solution to the advection–diffusion–reaction equation in unbounded domains is developed. The method is based on the concept of artificial boundary conditions (ABCs), and employs the techniques of time and dimensional splitting of the partial differential equation coupled with domain decomposition of the original infinite space. The essentials of the method is that it is applicable for solving a wide class of mass transportation problems in domain of drastically complex geometries, realisable from the computation standpoint, and provides a highly accurate solution at minimal computational efforts.

Keywords: Artificial (numerical) boundary conditions, advection–diffusion–reaction equation, splitting, domain decomposition.

Resumen

Se desarrolla un método para la solución numérica de la ecuación de advección–difusión–reacción en dominios infinitos. El método se basa en el concepto de condiciones de frontera artificiales (CFAs), y utiliza las técnicas de escisión del operador por tiempo y por espacio junto con la de descomposición de dominio para el espacio original infinito. Los esenciales del método son lo que es aplicable para dar solución a una amplia clase de los problemas de transporte de masa en dominios de la geometría demasiado compleja, realizable desde el punto de vista numérico, y además proporciona una alta precisión de la solución con mínimos esfuerzos computacionales.

Palabras clave: Condiciones de frontera artificiales (numéricas), ecuación de advección–difusión–reacción, escisión del operador, descomposición de dominio.

DESCARGAR ARTÍCULO EN FORMATO PDF

References

1. Bayliss and E. Turkel, Far Field Boundary Conditions for Compressible Flows, J. Comput. Phys. 48 (1982) 182–199.        [ Links ]

2. D.M. Filatov, On Two Computational Techniques of Constructing Boundary Conditions for the Diffusion Equation, in: H. Sossa et al., eds., Proc. of the 11^th International Conference on Computing (CIC'2002), National Polytechnic Inst, Mexico City, Mexico, November 25–29, 2002, pp. 399–404.        [ Links ]

3. D. M. Filatov, On Local Artificial Boundary Conditions for the Diffusion Equation in Case of 2D Convex Computational Domain, Appl. Numer. Math., submitted.        [ Links ]

4. D. M. Filatov, On Accurate Flexible Artificial Boundary Conditions for the Linear Advection–Diffusion–Reaction Equation, in: Proc. of the 15^th Annual International Conference on Domain Decomposition (DD15), Free Univ. of Berlin, Germany, 21–25 July, 2003, p. 45.        [ Links ]

5. D. M. Filatov, Splitting as an Approach to Constructing Local Exact Artificial Boundary Conditions, J. Comput. Appl. Math., submitted.        [ Links ]

6. D. M. Filatov, Method of Splitting as an Approach to Constructing Artificial Boundary Conditions, in: B. G. Mikhailenko et al., eds., Proc. of the International Conference on Mathematical Methods in Geophysics (MMG–2003), Inst. of Comput. Math, and Math. Geophys., Rus. Acad. Sci., Novosibirsk, Russia, October 8–12, 2003, pp. 685–689.        [ Links ]

7. D. Givoli, Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992.        [ Links ]

8. B. Gustafsson, The Choice of Numerical Boundary Conditions for Hyperbolic Systems, J. Comput. Phys. 48 (1982) 270–283.        [ Links ]

9. B. Gustafsson, Far–Field Boundary Conditions for Time–Dependent Hyperbolic Systems, SIAM J. Sci. Stat. Comput. 9 (1988) 812–828.        [ Links ]

10. L. Halpern, Artificial Boundary Conditions for the Linear Advection Diffusion Reaction Equation, Math. Comp. 46 (1986) 425–438.        [ Links ]

11. L. Halpern, Artificial Boundary Conditions for Incompletely Parabolic Perturbations of Hyperbolic Systems, SIAM J. Math. Anal. 22 (1991) 1256–1283.        [ Links ]

12. I. Herrera, New Approach to Advection–Dominated Flows and Comparison with Other Methods, in: Computational Mechanics, Vol. 2, Springer–Verlag, Heidelberg (1988).        [ Links ]

13. I. Herrera, M. A. Celia, and J. D. Martinez, Localized Adjoint Method as a New Approach to Advection–Dominated Flows, in: J. E. Moore et al., eds., Recent Advances in Ground–Water Hydrology. American Institute of Hydrology, (1989) 321–327.        [ Links ]

14. F. P. Incropera and D. P. De Witt, Fundamentals of Heat Transfer, John Wiley & Sons, 1996.        [ Links ]

15. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw–Hill, N. Y., 1968.        [ Links ]

16. G. I. Marchuk, Mathematical Models in Environmental Problems, Elsevier, 1986.        [ Links ]

17. W. H. Press et al., Numerical Recipes in C. The Art of Scientific Computing, Camb. Univ. Pr., 1992.        [ Links ]

18. S. V. Tsynkov, An Application of Non–Local External Conditions for Viscous Flow Computations, J. Comput. Phys. 116 (1995) 212–225.        [ Links ]

19. S. V. Tsynkov, Numerical Solution of Problems on Unbounded Domains. A Review, Appl. Numer. Math. 27 (1998) 465–532.        [ Links ]

20. S. V. Tsynkov, External Boundary Conditions for Three–Dimensional Problems of Computational Aerodynamics, SIAM J. Sci. Comput. 21 (1999) 166–206.        [ Links ]

21. M. N. Yudin, An Alternating Method for Numerical Solution to Geoelectrical Problems. Math. Meth. Geoelec, USSR Acad. Sci., IZMIRAN (199,2) 47–52.        [ Links ]

22. M. N. Yudin, On Integro–Differential Asymptotic Boundary Conditions in Primal Geoelectrical Problems. Izv. VUZov, Geol. & Prospecting 1 (1984) 97–101.        [ Links ]