Skip to main content
SpringerLink
Log in

Estimates of the hyperbolization effect on the heat equation

  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The difference between the solutions of the heat equation and its hyperbolized version is estimated. The estimates are obtained in the L 2 norm for the anisotropic heat equation and in the C norm for the one-dimensional case with constant coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. A. Davydov, B. N. Chetverushkin, and E. V. Shil’nikov, “Simulating flows of incompressible and weakly compressible fluids on multicore hybrid computer systems,” Comput. Math. Math. Phys. 50 (12), 2157–2165 (2010).

    Article  MathSciNet  Google Scholar 

  2. B. N. Chetverushkin and E. V. Shilnikov, “Flux relaxation as an approach to the stability improvement for explicit finite difference schemes,” CD Proceedings of the 2nd International Conference on Engineering Optimization (EngOpt-2010), Lisbon, Portugal, September 2010.

    Google Scholar 

  3. B. N. Chetverushkin, “Resolution limits of continuous media mode and their mathematical formulations,” Math. Model. Comput. Simul. 5 (3), 266–279 (2013).

    Article  MathSciNet  Google Scholar 

  4. B. N. Chetverushkin, D. N. Morozov, M. A. Trapeznikova, N. G. Churbanova, and E. V. Shil’nikov, “An explicit scheme for the solution of the filtration problems,” Math. Model. Comput. Simul. 2 (6), 669–677 (2010).

    Article  MathSciNet  Google Scholar 

  5. B. N. Chetverushkin and A. V. Gulin, “Explicit schemes and numerical simulation using ultrahigh-performance computer systems,” Dokl. Math. 86 (2), 681–683 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. N. Chetverushkin, E. V. Shilnikov, and A. A. Davydov, “Numerical simulation of continuous media problems on hybrid computer systems,” Adv. Eng. Software 60–61, 42–47 (2013); http://dx.doi.org/10.1016/j.advengsoft.2013.02.003.

    Article  Google Scholar 

  7. C. I. Repin and B. N. Chetverushkin, “Estimates of the difference between approximate solutions of the Cauchy problems for the parabolic diffusion equation and a hyperbolic equation with a small parameter,” Dokl. Math. 88 (1), 417–420 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. A. Davydov and E. V. Shilnikov, “Numerical simulation of the low compressible viscous gas flows on GPU-based hybrid supercomputers,” Parallel Computing: Accelerating Computational Science Engineering (CSE), Advances in Parallel Computing, Ed. by M. Bader, A. Bode, H.-J. Bungartz, M. Gerndt, G. R. Joubert, and F. Peters (IOS, 2014), Vol. 25, pp. 315–323.

    Google Scholar 

  9. M. A. Trapeznikova, N. G. Churbanova, A. A. Lyupa, and D. N. Morozov, “Simulation of multiphase flows in the subsurface on GPU-based supercomputers,” Parallel Computing: Accelerating Computational Science Engineering (CSE), Advances in Parallel Computing, Ed. by M. Bader, A. Bode, H.-J. Bungartz, M. Gerndt, G. R. Joubert, and F. Peters (IOS, 2014), Vol. 25, pp. 324–333.

    Google Scholar 

  10. E. V. Shil’nikov, Preprint No. 33, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2014).

  11. D. N. Morozov, M. A. Trapeznikova, B. N. Chetverushkin, and N. G. Churbanova, “Application of explicit schemes for the simulation of the two-phase filtration process,” Math. Model. Comput. Simul. 4 (1), 62–67 (2012).

    Article  MathSciNet  Google Scholar 

  12. N. V. Isupov, M. A. Trapeznikova, N. G. Churbanova, and E. V. Shil’nikov, “Modeling the infiltration of multiphase fluid flows in a layered porous medium,” Math. Model. Comput. Simul. 3 (1), 35–45 (2011).

    Article  Google Scholar 

  13. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1999; Dover, New York, 2011).

    Google Scholar 

  14. A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics: A Unified Introduction with Applications (Birkhäuser, Basel, 1987; Intellekt, Dolgoprudnyi, 2007).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

    E. E. Myshetskaya & V. F. Tishkin

Authors
  1. E. E. Myshetskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. F. Tishkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to E. E. Myshetskaya or V. F. Tishkin.

Additional information

Original Russian Text © E.E. Myshetskaya, V.F. Tishkin, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 8, pp. 1299–1304.

In blessed memory of Professor A.P. Favorskii

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Myshetskaya, E.E., Tishkin, V.F. Estimates of the hyperbolization effect on the heat equation. Comput. Math. and Math. Phys. 55, 1270–1275 (2015). https://doi.org/10.1134/S0965542515080138

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542515080138

Keywords

Search

Navigation