A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

G. I. Shishkin   ; L. P. Shishkina

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 12, pp. 2113-2133

Résumé

For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter $\varepsilon$, where $\varepsilon\in(0,1]$, the grid approximation of the Dirichlet problem on a rectangular domain in the $(x,t)$-plane is examined. For small $\varepsilon$, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of $\varepsilon$-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges $\varepsilon$-uniformly in the maximum norm at the rate of $O(N^{-2}\ln^2N+N_0^{-1})$, where $N+1$ and $N_0+1$ are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme converges $\varepsilon$-uniformly at the rate of $O(N^{-4}\ln^4N+N_0^{-2})$. For fixed values of the parameter, the convergence rate is $O(N^{-4}+N_0^{-2})$.

Bibliographie

[1] Schlichting H., Boundary layer Theory, 7th ed., McGraw-Hill, New York, 1979 | MR | Zbl

[2] Oleinik O. A., Samohin V. N., Mathematical models in boundary layer theory, Chapman Hall/CRC, Boca Raton, 1999 | MR | Zbl

[3] Dorodnitsyn A. A., Meller N. A., “O nekotorykh podkhodakh k resheniyu statsionarnykh uravnenii Nave–Stoksa”, Zh. vychisl. matem. i matem. fiz., 8:2 (1968), 393–402 | MR

[4] Kerimov M. K., “Pamyati Anatoliya Alekseevicha Dorodnitsyna”, Zh. vychisl. matem. i matem. fiz., 35:6 (1995), 819–842 | MR | Zbl

[5] Shishkin G. I., Shishkina L. P., Difference methods for singular perturbation problems, Monographs Surveys in Pure Appl. Math., Chapman and Hall/CRC, 2009 | MR | Zbl

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

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

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

[9] 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. Appl. Math., 8:2 (1955), 129–145 | DOI | MR | Zbl

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

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

[12] Linß T., “The necessity of Shishkin decompositions”, Appl. Math. Lett., 14:7 (2001), 891–896 | DOI | MR | Zbl

[13] Linß T., Stynes M., “Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem”, J. Math. Anal. Appl., 261:2 (2001), 604–632 | DOI | MR | Zbl

[14] O'Riordan E., Shishkin G. I., “Parameter uniform numerical methods for singularly perturbed elliptic problems with parabolic boundary layers”, Appl. Numer. Math., 58:12 (2008), 1761–1772 | DOI | MR

[15] Roos H.-G., Stynes M., Tobiska L., Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, 2nd ed., Springer, Berlin, 2008 | MR

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

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

[18] Böchmer K., Stetter H., Defect correction methods. Theory and applications, Comput. Suppl., 5, Springer, Wien, 1984 | MR

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

[20] Shishkin G. I., “Povyshenie tochnosti reshenii raznostnykh skhem dlya parabolicheskikh uravnenii s malym parametrom pri starshikh proizvodnykh”, Zh. vychisl. matem. i matem. fiz., 24:6 (1984), 864–875 | MR | Zbl

[21] Hemker P. W., Shishkin G. I., Shishkina L. P., “The use of defect correction for the solution of parabolic singular perturbation problems”, Z. angew. Math. Mech., 77:1 (1997), 59–74 | DOI | MR | Zbl

[22] 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

[23] Shishkin G. I., “Setochnye approksimatsii s uluchshennoi skorostyu skhodimosti dlya singulyarno vozmuschennykh ellipticheskikh uravnenii v oblastyakh s kharakteristicheskimi granitsami”, Sibirskii zhurnal vychisl. matem., 5:1 (2002), 71–92 | Zbl

[24] Comp. Math. Math. Phys., 44:2 (2004), 309–316 | MR | Zbl

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

[26] Shishkin G. I., “Robust novel high-order accurate numerical methods for singularly perturbed convection-diffusion problems”, Math. Model. Anal., 10:4 (2005), 393–412 | MR | Zbl

[27] Shishkin G. I., “Metod Richardsona povysheniya tochnosti reshenii singulyarno vozmuschennykh ellipticheskikh uravnenii s konvektsiei”, Izv. vuzov. Matematika, 2006, no. 2, 57–71 | MR | Zbl

[28] Shishkin G. I., “Skhema Richardsona dlya singulyarno vozmuschennogo parabolicheskogo uravneniya reaktsii-diffuzii s razryvnym nachalnym usloviem”, Zh. vychisl. matem. i matem. fiz., 49:8 (2009), 1416–1436 | MR | Zbl

[29] Shishkin G. I., Shishkina L. P., “Skhema Richardsona povyshennogo poryadka tochnosti dlya semilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Zh. vychisl. matem. i matem. fiz., 50:3 (2010), 458–478 | MR

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

[31] Shishkin G. I., “Setochnaya approksimatsiya s uluchshennoi skorostyu skhodimosti dlya singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Tr. IMM UrO RAN, 9, no. 1, Ekaterinburg, 2003, 165–182

[32] Bakhvalov N. S., Chislennye metody, Nauka, M., 1973 | MR | Zbl

[33] Doolan E. P., Miller J. J. H., Shilders W. H. A., Uniform numerical methods for problem with initial boundary layers, Boole Press, Dublin, 1980 | MR | Zbl

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