| Title:
|
An efficient linear numerical scheme for the Stefan problem, the porous medium equation and nonlinear cross-diffusion systems (English) |
| Author:
|
Molati, Motlatsi |
| Author:
|
Murakawa, Hideki |
| Language:
|
English |
| Journal:
|
Proceedings of Equadiff 14 |
| Volume:
|
Conference on Differential Equations and Their Applications, Bratislava, July 24-28, 2017 |
| Issue:
|
2017 |
| Year:
|
|
| Pages:
|
305-314 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Stefan problem, Porous medium equation, Cross-diffusion system, Degenerate convection-reaction-diffusion equation, Linear scheme, Error estimate, Numerical method |
| MSC:
|
35K55 |
| MSC:
|
65M12 |
| MSC:
|
80A22 |
| MSC:
|
92D25 |
| . |
| Date available:
|
2019-09-27T08:15:42Z |
| Last updated:
|
2019-09-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/703037 |
| . |
| Reference:
|
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| Reference:
|
[2] Berger, A.E., Brezis, H., Rogers, J.C.W.: A numerical method for solving the problem $ut − \Delta f(u) = 0$., R.A.I.R.O. Anal. Numér., 13 (1979), pp. 297–312. MR 0555381 |
| Reference:
|
[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. MR 0921832, 10.1051/m2an/1987210406551 |
| Reference:
|
[4] Molati, M., Murakawa, H.: Exact solutions of nonlinear diffusion-convection-reaction equation: A Lie symmetry analysis approach., preprint. MR 3854261 |
| Reference:
|
[5] Murakawa, H.: A linear scheme to approximate nonlinear cross-diffusion systems., Math. Mod. Numer. Anal., 45 (2011), pp. 1141–1161. MR 2833176, 10.1051/m2an/2011010 |
| Reference:
|
[6] Murakawa, H.: Error estimates for discrete-time approximations of nonlinear cross-diffusion systems., SIAM J. Numer. Anal., 52(2) (2014), pp. 955–974. MR 3196950, 10.1137/130911019 |
| Reference:
|
[7] Murakawa, H.: A linear finite volume method for nonlinear cross-diffusion systems., Numer. Math., 136(1) (2017), pp. 1–26. MR 3632917, 10.1007/s00211-016-0832-z |
| Reference:
|
[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. MR 3768238, 10.1007/s13160-017-0279-3 |
| Reference:
|
[9] Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species., J. Theor. Biol., 79 (1979), pp. 83–99. MR 0540951, 10.1016/0022-5193(79)90258-3 |
| . |
