An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems

Molati, Motlatsi; Murakawa, Hideki

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 305-314

Abstract

top
This paper deals with nonlinear diffusion problems which include the Stefan problem, the porous medium equation and cross-diffusion systems. We provide a linear scheme for these nonlinear diffusion problems. The proposed numerical scheme has many advantages. Namely, the implementation is very easy and the ensuing linear algebraic systems are symmetric, which show low computational cost. Moreover, this scheme has the accuracy comparable to that of the wellstudied nonlinear schemes and make it possible to realize the much faster computation rather than the nonlinear schemes with the same level of accuracy. In this paper, numerical experiments are carried out to demonstrate efficiency of the proposed scheme.

How to cite

top

Molati, Motlatsi, and Murakawa, Hideki. "An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 305-314. <http://eudml.org/doc/294897>.

@inProceedings{Molati2017,
abstract = {This paper deals with nonlinear diffusion problems which include the Stefan problem, the porous medium equation and cross-diffusion systems. We provide a linear scheme for these nonlinear diffusion problems. The proposed numerical scheme has many advantages. Namely, the implementation is very easy and the ensuing linear algebraic systems are symmetric, which show low computational cost. Moreover, this scheme has the accuracy comparable to that of the wellstudied nonlinear schemes and make it possible to realize the much faster computation rather than the nonlinear schemes with the same level of accuracy. In this paper, numerical experiments are carried out to demonstrate efficiency of the proposed scheme.},
author = {Molati, Motlatsi, Murakawa, Hideki},
booktitle = {Proceedings of Equadiff 14},
keywords = {Stefan problem, Porous medium equation, Cross-diffusion system, Degenerate convection-reaction-diffusion equation, Linear scheme, Error estimate, Numerical method},
location = {Bratislava},
pages = {305-314},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems},
url = {http://eudml.org/doc/294897},
year = {2017},
}

TY - CLSWK
AU - Molati, Motlatsi
AU - Murakawa, Hideki
TI - An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 305
EP - 314
AB - This paper deals with nonlinear diffusion problems which include the Stefan problem, the porous medium equation and cross-diffusion systems. We provide a linear scheme for these nonlinear diffusion problems. The proposed numerical scheme has many advantages. Namely, the implementation is very easy and the ensuing linear algebraic systems are symmetric, which show low computational cost. Moreover, this scheme has the accuracy comparable to that of the wellstudied nonlinear schemes and make it possible to realize the much faster computation rather than the nonlinear schemes with the same level of accuracy. In this paper, numerical experiments are carried out to demonstrate efficiency of the proposed scheme.
KW - Stefan problem, Porous medium equation, Cross-diffusion system, Degenerate convection-reaction-diffusion equation, Linear scheme, Error estimate, Numerical method
UR - http://eudml.org/doc/294897
ER -

References

top
  1. Barenblatt, G.I., On some unsteady motion of a liquid or a gas in a porous medium, , Prikl. Math. Meh., 16 (1952), pp. 67–78. MR0046217
  2. Berger, A.E., Brezis, H., Rogers, J.C.W., A numerical method for solving the problem u t − Δ f ( u ) = 0 , , R.A.I.R.O. Anal. Numér., 13 (1979), pp. 297–312. MR0555381
  3. Magenes, E., Nochetto, R.H., Verdi, C., Energy error estimates for a linear scheme to approximate nonlinear parabolic problems, , Math. Mod. Numer. Anal., 21 (1987), pp. 655–678. MR0921832
  4. Molati, M., Murakawa, H., Exact solutions of nonlinear diffusion-convection-reaction equation: A Lie symmetry analysis approach, , preprint. MR3854261
  5. Murakawa, H., A linear scheme to approximate nonlinear cross-diffusion systems, , Math. Mod. Numer. Anal., 45 (2011), pp. 1141–1161. MR2833176
  6. Murakawa, H., Error estimates for discrete-time approximations of nonlinear cross-diffusion systems, , SIAM J. Numer. Anal., 52(2) (2014), pp. 955–974. MR3196950
  7. Murakawa, H., A linear finite volume method for nonlinear cross-diffusion systems, , Numer. Math., 136(1) (2017), pp. 1–26. MR3632917
  8. Murakawa, H., An efficient linear scheme to approximate nonlinear diffusion problems, , to appear in Jpn. J. Ind. Appl. Math., DOI: 10.1007/s13160-017-0279-3. MR3768238
  9. Shigesada, N., Kawasaki, K., Teramoto, E., Spatial segregation of interacting species, , J. Theor. Biol., 79 (1979), pp. 83–99. MR0540951

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.