NAA 2004: Numerical Analysis and Its Applications pp 352-359 | Cite as
Comparison of a Rothe-Two Grig Method and Other Numerical Schemes for Solving Semilinear Parabolic Equations
Conference paper
Abstract
A technique combined the Rothe method with two-grid (coarse and fine) algorithm of Xu [18] for computation of numerical solutions of nonlinear parabolic problems with various boundary conditions is presented. For blow-up solutions we use a decreasing variable step in time, according to the growth of the solution. We give theoretical results, concerning convergence of the numerical solutions to the analytical ones. Numerical experiments for comparison the accuracy of the algorithm with other known numerical schemes are discussed.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Andreuci, D., Gianni, R.: Global existence and blow up in a parabolic problem with nonlocal dynamical boundary conditions. Adv. Diff. Equations 1, 5, 729–752 (1996)MATHGoogle Scholar
- 2.Cimrák, I.: Numerical solution of degenerate convection-diffusion problem using broyden scheme. In: Proceedings of ALGORITMY, Conf. on Sci. Comput., pp. 14–22 (2002)Google Scholar
- 3.Courant, R., Hilbert, D.: Methoden der Mathematishen Physika. Springer, Berlin (1965)Google Scholar
- 4.Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1973)Google Scholar
- 5.Escher, J.: On the quasilinear behaviour of some semilinear parabolic problems. Diff. Integral Eq. 8, 247–267 (1995)MATHMathSciNetGoogle Scholar
- 6.Fila, M., Quittner, P.: Global solutions of the Laplace equation with a nonlinear dynamical boundary condition. Math. Appl. Sci. 20, 1325–1333 (1997)MATHCrossRefMathSciNetGoogle Scholar
- 7.Fila, M., Quittner, P.: Large time behavior of solutions of semilinear parabolic equations with nonlinear dynamical boundary conditions. Topics in Nonlinear Analysis, Progr. Nonl. Diff. Eq. Appl. 35, 251–272 (1999)MathSciNetGoogle Scholar
- 8.Gröger, K.: Initial boundary value problems from semiconductor device theory. ZAMM 67, 345–355 (1987)MATHCrossRefGoogle Scholar
- 9.Kačur, J.: Nonlinear parabolic equations with mixed nonlinear and nonstationary boundary conditions. Math. Slovaca 30, 213–237 (1990)Google Scholar
- 10.Kirane, M.: Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type. Hokkaido Math. J. 21, 221–229 (1992)MATHMathSciNetGoogle Scholar
- 11.Koleva, M., Vulkov, L.: On the blow-up of finite difference solutions to the heat-diffusion equation with semilinear dynamical boundary conditions. Appl. Math. & Comp. (in press)Google Scholar
- 12.Ladyzhenskaya, A., Solnnikov, A., Uraltseva, N.: Linear and Quasi-Linear Equations of Parabolic Type. In: Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)Google Scholar
- 13.Lapidus, L., Amundson, R.N.: Chemical Reactor Theory. Prentice-Hall, Englewood Cliffs (1977)Google Scholar
- 14.Nakagava, T.: Blowing up of a finite difference solution to \(u_{t}=u_{xx}+u^{2}\). Appl. Math. & Optimization 2, 337–350 (1976)CrossRefGoogle Scholar
- 15.Samarskii, A., Galakttionov, V., Kurdyumov, M.A.: Blow-up in problems for quasilinear parabolic equations (Walter de Gruyter Trans., Expositionin Mathematics) Nauka, Moskow, 19 (Original work publushed in 1987) Berlin (1995) (in Russian)Google Scholar
- 16.Velazquez, L.: Blow-up for semilinear parabolic equations. In: Herrero, M.A., Zuazua, E. (eds.) Recent Adv. in PDEs, pp. 131–145. John Wiley and sons, Chichester (1993)Google Scholar
- 17.Vold, R.D., Vold, M.J.: Colloid and Interface Chemistry. Addision - Wesley, Reading (1983)Google Scholar
- 18.Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15(1), 231–237 (1994)MATHCrossRefMathSciNetGoogle Scholar
Copyright information
© Springer-Verlag Berlin Heidelberg 2005