Grid approximation of a singularly perturbed quasilinear parabolic convection-diffusion equation on a priori adapted meshes

G. I. Shishkin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 149 (2007) no. 4, pp. 146-172

Résumé

An initial-boundary value problem is considered for a quasilinear singularly perturbed parabolic convection-diffusion equation. For such a problem, a solution of a classical difference scheme on uniform grid converges at the rate $\mathcal O((\varepsilon+N^{-1})^{-1}N^{-1}+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of nodes in the meshes in $x$ and $t$ respectively; the scheme converges only under the condition $N^{-1}\ll\varepsilon$. In the present paper, nonlinear and linearized finite difference schemes are constructed on a priori sequentially adapted grids, and their convergence is studied. The construction of the schemes is carried out on the basis of a majorant to the singular component of the discrete solution on uniform grids that allows us to find a priori subdomains where the computed solution requires a further improvement. Such subdomain is defined by the perturbation parameter $\varepsilon$, the step-size of a uniform mesh in $x$, and also by the required accuracy of the grid solution and the prescribed number $K$ of iterations to refine the solution. The advantage of this approach consists in the uniform meshes used. The error of the discrete solution depends weakly on the parameter $\varepsilon$. The schemes that are constructed in the iterative process converge almost $\varepsilon$-uniformly, namely, under the condition $N^{-1}\ll\varepsilon^{\nu}$, where the value $\nu=\nu(K)$ can be chosen arbitrarily small for sufficiently large $K$.

Bibliographie

[1] Bakhvalov N. C., “K optimizatsii metodov resheniya kraevykh zadach pri nalichii pogranichnogo sloya”, Zhurn. vychisl. matem. i matem. fiz., 9:4 (1969), 841–859 | Zbl

[2] Ilin A. M., “Raznostnaya skhema dlya differentsialnogo uravneniya s malym parametrom pri starshei proizvodnoi”, Matem. zametki, 6:2 (1969), 237–248

[3] Dulan E., Miller Dzh., Shilders U., Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem, Mir, M., 1983, 199 pp. | MR

[4] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, Izd-vo UrO RAN, Ekaterinburg, 1992, 232 pp.

[5] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific Publishing Co., Singapore, 1996, 180 pp. | MR

[6] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E., Shishkin G. I., CRC Press, Boca Raton, 2000, 270 pp. | Zbl

[7] Roos H.-G., Stynes M., Tobiska L., Numerical Methods for Singularly Perturbed Differential Equations, Springer, Heidelberg, 1996, 364 pp. | MR

[8] Shishkin G. I., “Approksimatsiya reshenii singulyarno vozmuschennykh kraevykh zadach s parabolicheskim pogranichnym sloem”, Zhurn. vychisl. matem. i matem. fiz., 29:7 (1989), 963–977 | MR

[9] Hemker P. W., Shishkin G. I., “On a class of singularly perturbed boundary value problems for which an adaptive mesh technique is necessary”, Proc. of the Second Internat. Colloquium on Numerical Analysis, eds. D. Bainov, V. Covachev, International Science Publishers, 1994, 83–92 | MR | Zbl

[10] Shishkin G. I., “On finite difference fitted schemes for singularly perturbed boundary value problems with a parabolic boundary layer”, J. Math. Anal. and Applications, 208 (1997), 181–204 | DOI | MR | Zbl

[11] Farrell P. A., Miller J. J. H., O'Riordan E., Shishkin G. I., “On the non-existence of e-uniform finite difference methods on uniform meshes for semilinear two-point boundary value problems”, Math. Comp., 67:222 (1998), 603–617 | DOI | MR | Zbl

[12] Shishkin G. I., “Aposteriorno adaptiruemye (po gradientu resheniya) setki v approksimatsii singulyarno vozmuschennykh uravnenii konvektsii-diffuzii”, Vychisl. tekhnologii, 6:1–2 (2001), 72–87 | MR | Zbl

[13] Shishkin G. I., “Approksimatsiya singulyarno vozmuschennykh uravnenii reaktsii-diffuzii na adaptivnykh setkakh”, Matem. modelirovanie, 13:3 (2001), 103–118 | MR | Zbl

[14] Shishkin G. I., Shishkina L. P., Hemker P. W., “A class of singularly perturbed convection-diffusion problems with a moving interior layer, $A~Posteriori$ Adaptive Mesh Technique”, Computational Methods in Applied Mathematics, 4:1 (2004), 105–127 | MR | Zbl

[15] Shishkin G. I., “Ispolzovanie reshenii na vlozhennykh setkakh pri approksimatsii singulyarno vozmuschennogo parabolicheskogo uravneniya konvektsii–diffuzii na adaptiruyuschikhsya setkakh”, Zhurn. vychisl. matem. i matem. fiz., 46:9 (2006), 1617–1637 | MR

[16] Shishkin G. I., “A finite difference scheme on apriori adapted meshes for a singularly perturbed parabolic convection-diffusion problem equation”, accepted, Numer. Math. J. Chinesse Univ.

[17] Ladyzhenskaya O. A., Colonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR

[18] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989, 614 pp. | MR

[19] Samarskii A. A., Nikolaev E. S., Metody resheniya setochnykh uravnenii, Nauka, M., 1978, 592 pp. | MR

[20] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989, 608 pp. | MR

[21] Marchuk G. I., Shaidurov V. V., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979, 320 pp. | MR

[22] Bakhvalov N. S., Chislennye metody, Nauka, M., 1973, 631 pp. | MR | Zbl

[23] Hemker P. W., Shishkin G. I., Shishkina L. P., “$\varepsilon$-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems”, IMA J. Numer. Anal., 20:1 (2000), 99–121 | DOI | MR | Zbl

[24] Hemker P. W., Shishkin G. I., Shishkina L. P., “Novel defect-correction high-order, in space and time, accurate schemes for parabolic singularly perturbed convection-diffusion problems”, Comp. Methods in Appl. Math., 3:3 (2003), 387–404 | MR | Zbl

[25] Shishkin G. I., Shishkina L. P., “Metod Richardsona vysokogo poryadka tochnosti dlya kvazilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya reaktsii-diffuzii”, Differents. uravneniya, 41:7 (2005), 980–989 | MR

[26] Shishkin G. I., Shishkina L. P., “The Richardson extrapolation technique for quasilinear parabolic singularly perturbed convection-diffusion equations”, Journal of Physics. Conference Series, 55, International Workshop on Multi-Rate Processes and Hysteresis, (3–8 April 2006, University College Cork, Ireland) (2006), 203–213 ; Режим доступа: http://www.iop.org/EJ/toc/1742-6596/55/1/ | DOI

[27] Wesseling P., Principles of Computational Fluid Dynamics, Springer-Verlag, Berlin, 2001, 652 pp. | MR