Blow-up phenomena in the model of a space charge stratification in semiconductors: numerical analysis of original equation reduction to a differential-algebraic system

Authors

DOI:

https://doi.org/10.26089/NumMet.v17r440

Keywords:

numerical diagnostics of solution’s blow-up, Rosenbrock method, partial differential equations, differential-algebraic equations

Abstract

The efficiency of one of the methods for the numerical diagnostics of solution’s blow-up is shown using the example of solving a nonlinear Sobolev-type equation that describes a space charge stratification in semiconductors. An approach to reduce the original partial differential equation to a differential-algebraic system is used. This system is solved by the Rosenbrock scheme with a complex coefficient. The numerical diagnostics of solution’s blow-up is based on the Richardson extrapolation procedure.

Author Biographies

D.V. Lukyanenko

A.A. Panin

References

  1. E. Mitidieri and S. I. Pokhozhaev, “A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 234, 3-383 (2001) [Proc. Steklov Inst. Math. 234, 1-362 (2001)].
  2. H. A. Levine, “Some Nonexistence and Instability Theorems for Solutions of Formally Parabolic Equations of the Form P𝓊𝓉 = -A𝓊 + ℱ(𝓊),” Arch. Rational Mech. Anal. 51 (5), 371-386 (1973).
  3. H. A. Levine, “Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form P𝓊𝓉𝓉 = -A𝓊 + ℱ(𝓊),” Trans. Am. Math. Soc. 192, 1-21 (1974).
  4. V. K. Kalantarov and O. A. Ladyzhenskaya, “The Occurrence of Collapse for Quasilinear Equations of Parabolic and Hyperbolic Types,” Zap. Nauch. Semin. Leningr. Otd. Mat. Inst. Steklova 69, 77-102 (1977). [J. Math. Sci. 10 (1), 53-70 (1978)].
  5. A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner, Linear and Nonlinear Equations of Sobolev Type (Fizmatlit, Moscow, 2007) [in Russian].
  6. M. O. Korpusov, Blow-up in Nonclassical Wave Equations (Editorial, Moscow, 2010) [in Russian].
  7. M. O. Korpusov, “Blow-up of Ion Acoustic Waves in a Plasma,” Mat. Sb. 202 (1), 37-64 (2011) [Sb. Math. 202 (1), 35-60 (2011)].
  8. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations (Nauka, Moscow, 1987; Gruyter, Berlin, 1995).
  9. V. A. Galaktionov and S. I. Pokhozhaev, “Third-Order Nonlinear Dispersive Equations: Shocks, Rarefaction, and Blowup Waves,” Zh. Vychisl. Mat. Mat. Fiz. 48 (10), 1819-1846 (2008) [Comput. Math. Math. Phys. 48 (10), 1784-1810 (2008)].
  10. D. E. Pelinovsky and C. Xu, “On Numerical Modelling and the Blow-up Behavior of Contact Lines with a 180° Contact Angle,” J. Eng. Math. 92, 31-44 (2015).
  11. A. Cangiani, E. H. Georgoulis, I. Kyza, and S. Metcalfe, “Adaptivity and Blow-up Detection for Nonlinear Evolution Problems,” arXiv preprint: 1502.03250v1 [math.NA] (Cornell Univ. Library, Ithaca, 2015), available at
    https://arxiv.org/abs/1502.03250/.
  12. R. Haynes and C. Turner, “A Numerical and Theoretical Study of Blow-up for a System of Ordinary Differential Equations Using the Sundman Transformation,” Atl. Electron. J. Math. 2 (1), 1-13 (2007).
  13. M. Berger and R. V. Kohn, “A Rescaling Algorithm for the Numerical Calculation of Blowing-up Solutions,” Commun. Pure Appl. Math. 41 (6), 841-863 (1988).
  14. C.-H. Cho, “Numerical Detection of Blow-up: A New Sufficient Condition for Blow-up,” Japan J. Indust. Appl. Math. 33 (1), 81-98.
  15. E. A. Alshina, N. N. Kalitkin, and P. V. Koryakin, “Diagnostics of Singularities of Exact Solutions in Computations with Error Control,” Zh. Vychisl. Mat. Mat. Fiz. 45 (10), 1837-1847 (2005) [Comput. Math. Math. Phys. 45 (10), 1769-1779 (2005)].
  16. N. N. Kalitkin, A. B. Al’shin, E. A. Al’shina, and B. V. Rogov, Calculations on Quasi-Uniform Grids (Fizmatlit, Moscow, 2005) [in Russian].
  17. A. B. Al’shin and E. A. Al’shina, “Numerical Diagnosis of Blow-up of Solutions of Pseudoparabolic Equations,” J. Math. Sci. 148 (1), 143-162 (2008).
  18. J. Hoffman and C. Johnson, “Blow up of Incompressible Euler Solutions,” BIT Numer. Math. 48 (2), 285-307 (2008).
  19. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up Phenomena in the Model of a Space Charge Stratification in Semiconductors: Analytical and Numerical Analysis,” Math. Meth. Appl. Sci., 2016 (in press). doi: 10.1002/mma.4142
  20. M. O. Korpusov, D. V. Lukyanenko, A. A. Panin, and E. V. Yushkov, “Blow-up for One Sobolev Problem: Theoretical Approach and Numerical Analysis,” J. Math. Anal. Appl. 442 (2), 451-468 (2016).
  21. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1981; Dover, New York, 1999).
  22. E. Hairer and G. Wanner, Solving Ordinary Differential Equations. Stiff and Differential-Algebraic Problems (Springer, Berlin, 2002).
  23. N. N. Kalitkin, “Numerical Methods for Solving Stiff Systems,” Mat. Model. 7 (5), 8-11 (1995).
  24. H. H. Rosenbrock, “Some General Implicit Processes for the Numerical Solution of Differential Equations,” Comput. J. 5 (4), 329-330 (1963).
  25. A. B. Al’shin, E. A. Al’shina, N. N. Kalitkin, and A. B. Koryagina, “Rosenbrock Schemes with Complex Coefficients for Stiff and Differential Algebraic Systems,” Zh. Vychisl. Mat. Mat. Fiz. 46 (8), 1392-1414 (2006) [Comput. Math. Math. Phys. 46 (8), 1320-1340 (2006)].

Published

2016-10-16

How to Cite

Лукьяненко Д.В., Панин А.А. Blow-up Phenomena in the Model of a Space Charge Stratification in Semiconductors: Numerical Analysis of Original Equation Reduction to a Differential-Algebraic System // Numerical methods and programming. 2016. 17. 437-446. doi 10.26089/NumMet.v17r440

Issue

Section

Section 1. Numerical methods and applications

Most read articles by the same author(s)

1 2 > >>