Difference scheme of highest accuracy order for a singularly perturbed reaction-diffusion equation based on the solution decomposition method

G. I. Shishkin ; L. P. Shishkina

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 1, pp. 280-293

Résumé

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes whose solutions converge in the maximum norm uniformly with respect to the perturbation parameter $\varepsilon$, $\varepsilon \in (0,1]$ (i.e., $\varepsilon$-uniformly) with order of accuracy significantly greater than the achievable accuracy order for the Richardson method on piecewise-uniform grids. Important in this approach is the use of uniform grids for solving grid subproblems for regular and singular components of the grid solution. Using the asymptotic construction technique, a basic difference scheme of the solution decomposition method is constructed that converges $\varepsilon$-uniformly in the maximum norm at the rate ${\mathcal O} \left(N^{-2} \ln^2 N\right)$, where $N+1$ is the number of nodes in the uniform grids used. The Richardson extrapolation technique on three embedded grids is applied to the basic scheme of the solution decomposition method. As a result, we have constructed the Richardson scheme of the solution decomposition method with highest accuracy order. The solution of this scheme converges $\varepsilon$-uniformly in the maximum norm at the rate ${\mathcal O} \left(N^{-6} \ln^6 N\right)$.

Détail

Keywords: ; singularly perturbed boundary value problem; ordinary differential reaction-diffusion equation; decomposition of a discrete solution; asymptotic construction technique; difference scheme of the solution decomposition method; uniform grids; $\varepsilon$-uniform convergence; maximum norm; Richardson extrapolation technique; difference scheme of highest accuracy order.

Bibliographie

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

[2] Khemker P.V., Shishkin G.I., Shishkina L.P., “Dekompozitsiya metoda Richardsona vysokogo poryadka tochnosti dlya singulyarno vozmuschennogo ellipticheskogo uravneniya reaktsii-diffuzii”, Zhurn. vychisl. matematki i mat. fiziki, 44:2 (2004), 328–336 | MR

[3] Shishkina L.P., “The Richardson method of high-order accuracy in $t$ for a semilinear singularly perturbed parabolic reaction-diffusion equation on a strip”, Proc. of the internat. conf. on comput. math. (ICCM'2004), Part 2, ICM Publisher, Novosibirsk, 2004, 927–931

[4] 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 | Zbl

[5] Shishkin G.I., “Robust novel high-order accurate numerical methods for Singularly perturbed convection-diffusion problems”, Math. Modelling and Analysis, 10:4 (2005), 393–412 | MR | Zbl

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

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

[8] Shishkin G.I., Shishkina L.P., “Skhema Richardsona povyshennogo poryadka tochnosti dlya semilineinogo singulyarno vozmuschennogo ellipticheskogo uravneniya konvektsii-diffuzii”, Zhurn. vychisl. matematiki i mat. fiziki, 50:3 (2010), 458–478 | MR | Zbl

[9] Shishkin G.I., Shishkina L.P., Difference methods for singular perturbation problems, Ser. Monographs Surveys in Pure Applied Math, 140, Chapman HAll/CRC, Boca Raton, 2009, 408 pp. | MR | Zbl

[10] Shishkin G.I., Shishkina L.P., “Uluchshennaya raznostnaya skhema metoda dekompozitsii resheniya dlya singulyarno vozmuschennogo (obyknovennogo differentsialnogo) uravneniya reaktsii-diffuzii”, Tr. Instituta matematiki i mekhaniki UrO RAN, 16:1 (2010), 255–271

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

[12] 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

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

[14] Miller J.J.H., O'Riordan E., “Necessity of fitted operators and Shishkin meshes for resolving thin layer phenomena”, CWI Quarterly, 10:34 (1997), 207–213 | MR | Zbl

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

[16] Samarskii A.A., Teoriya raznostnykh skhem, Nauka, M., 1989, 616 pp. | MR

[17] Shishkin G.I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, Izd-vo UrO RAN, Ekaterinburg, 1992, 233 pp.

[18] Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M., Chislennye metody, Laboratoriya Bazovykh Znanii, M., 2001, 632 pp.