Necessary conditions for $\varepsilon$-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 10, pp. 1706-1726

Résumé

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis $x$ in the positive direction. For small values of the parameter $\varepsilon$ (this is the coefficient of the higher order derivatives of the equation, $\varepsilon\in(0,1]$), a moving boundary layer appears in a neighborhood of the left lateral boundary $S_1^L$. In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N+N_0)$, where $N$ and $N_0$ define the number of mesh points in $x$ and $t$. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition $N^{-1}+N_0^{-1}\ll\varepsilon$. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of $S_1^L$ with respect to $x$ and $t$, the convergence under the condition $N^{-1}+N_0^{-1}\le\varepsilon^{1/2}$ cannot be achieved. Examination of widths that are similar to Kolmogorov's widths makes it possible to establish necessary and sufficient conditions for the $\varepsilon$-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges $\varepsilon$-uniformly at a rate of $O(N^{-1}\ln N+N_0)$.

Bibliographie

[1] Allen D. N., Southwell R. V., “Relaxation methods applied to determine the motion, in two dimensions, of viscous fluid past a fixed cylinder”, Quart. J. Mech. and Appl. Math., 8 (1955), 129–145 | DOI | MR | Zbl

[2] Bakhvalov H. S., “K optimizatsii metodov resheniya kraevykh zadam pri nalichii pogranichnogo sloya”, Zh. vychisl. matem. i matem. fiz., 9:4 (1969), 841–859 | Zbl

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

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

[5] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[6] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted numerical methods for singular perturbation problems, World Scient., Singapore, 1996 | MR

[7] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E., Shishkin G. I., Robust computational techniques for boundary layers, Chapman Hall/CRC Press, Boca Raton, 2000 | MR | Zbl

[8] Roos H.-G., Stynes M., Tobiska L., Numerical methods for singularly perturbed differential equations. Convectiondiffusion and flow problems, Springer, Berlin, 1996 | MR

[9] Eiseikin V. D., Grid generation methods, Springer, Berlin, 1999 | MR

[10] Shishkin G. I., “On numerical methods on adaptive meshes for a singularly perturbed reaction-diffusion equation with a moving concentrated source”, Finite Difference Schemes: Theory and Applic., Proc. Conf. FDS2000, Inst. Maths Inf., Vilnus, 2000, 205–214 | MR | Zbl

[11] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennogo parabolicheskogo uravneniya na sostavnoi oblasti v sluchae sosredotochennogo istochnika na dvizhuscheisya granitse razdela”, Zh. vychisl. matem. i matem. fiz., 43:12 (2003), 1806–1824 | MR | Zbl

[12] Shishkin G. I., Shishkina L. P., Hemker P. W., “A class of singularly perturbed convection-diffusion problems with a moving interior layer. An a posteriori adaptive mesh technique”, Comput. Meth. Appl. Math., 4:1 (2004), 105–127 | MR | Zbl

[13] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva H. H., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[14] Hemker P. W., Shishkin G. I., Shishkina L. P., “High-order time-accuracy schemes for parabolic singular perturbation problems with convection”, Rus. J. Numer. Analys. and Math. Modelling, 17:1 (2002), 1–24 | MR

[15] 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”, Comput. Meth. Appl. Math., 3:3 (2003), 387–404 | MR | Zbl

[16] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[17] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989 | MR

[18] Babenko K. I., Osnovy chislennogo analiza, Nauka, M., 1986 | MR