The immersed interface method for a nonlinear chemical diffusion equation with local sites of reactions

Numerical Algorithms

Volumn 36, Issue 4, 2004, Pages 285-307

  Kandilarov, Juri D ^a   Vulkov, Lubin G ^a  

^a UNIVERSITY OF RUSE   (Bulgaria)

Author keywords

Delta dirac function; Finite difference schemes; Immersed interface method; Nonlinear parabolic problems

Indexed keywords

EID: 26844460567     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11075-004-2170-y     Document Type: Article

References (24)

1. J.H. Ahlbery and E.H. Nilson, Convergence properties of the spline fit, J. SIAM 11 (1963) 95-104.
2. N.S.Bakhvalov, Numerical Methods (Nauka, Moscow, 1973).
3. R.P. Beyer and R.J. Leveque, Analysis of a one-dimensional model for the immersed boundary method, SIAM J. Numer. Anal. 29 (1992) 332-364.
4. K. Bimpong-Bota, A. Nizan, P. Ortoleva and J. Ross, Cooperative phenomena in analysis of catalytic sites, J. Chem. Phys. 66 (1970) 3650-3678.
5. I.A. Braianov and J.D. Kandilarov, On the realization and stability of difference schemes for diffusion problems with localized reactions, in: Notes on Numerical Fluid Mechanics, Vol. 62 (Vieweg, Braunschweig, 1997) pp. 201-209.
6. J.M. Chadam, A. Pierce and H.M. Yin, The blowup property of solution to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169(2) (1992) 313-327.
7. J.M. Chadam and H.M. Yin, A diffusion equation with localized chemical reactions, Proc. Edinburgh Math. Soc. 37 (1993) 101-118.
8. J.D. Kandilarov, A second-order difference method for solution of diffusion problems with localized chemical reactions, in: Proc. of the 2nd Internat. Conf. on Finite Difference Methods: Theory and Applications, Vol. 2, eds. A.A. Samarskii, P.N. Vabishchevich and P.P. Matus (Minsk, 1998) pp. 63-67.
9. J.D. Kandilarov, The immersed interface method for a reaction diffusion equation with a moving own concentrated source, in: Lecture Notes in Computer Science, Vol. 2542 (2003) pp. 506-513.
10. O.A. Ladyzhenskaja, V.A. Solonnikov and N.N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type (Nauka, Moscow, 1967) (in Russian).
11. R.J. Leveque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 31 (1994) 1019-1044.
12. Z. Li, The immersed interface method - a numerical approach to partial differential equations with interfaces, Ph.D. thesis, University of Washington, Seatle (1994).
13. Z. Li, Immersed interface method for moving interface problems, Numer. Algorithms 14 (1997) 269-294.
14. Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, North Carolina State University, NCSU CRSC-TR99-5 (1999).
15. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
16. A.P. Pierce and H. Rabitz, An analysis of the effect of defect structures on catalytic surfaces by the boundary element technique, Surface Science 202 (1988) 1-31.
17. A.A. Samarskii, Theory of Difference Schemes (Nauka, Moscow, 1989) (in Russian).
18. A.A. Samarskii and E.S. Nikolaev, Methods for Solving Difference Schemes (Nauka, Moscow, 1978) (in Russian).
19. J.M. Sanz-Serna and I. Abia, Interpolation of the coefficients in nonlinear elliptic Galerkin procedures, SIAM J. Numer. Anal. 21 (1984) 77-83.
20. VS. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1988) (in Russian).
21. R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1962).
22. L.G. Vulkov and J.D. Kandilarov, Construction and implementation of finite-difference schemes for systems of diffusion equations with localized nonlinear chemical reactions, Comput. Math. Math. Phys. 40 (2000) 705-717.
23. A. Wiegmann and K.P. Bube, The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal. 35 (1998) 177-201.
24. A. Wiegmann and K.P. Bube, The explicit jump immersed interface method: Finite difference methods for PDE with piecewise smoth solutions, SIAM J. Numer. Anal. 37 (2000) 827-862.