Информация о публикации

Просмотр записей
Инд. авторы: Khakimzyanov G., Dutykh D.
Заглавие: On supraconvergence phenomenon for second order centered finite differences on non-uniform grids
Библ. ссылка: Khakimzyanov G., Dutykh D. On supraconvergence phenomenon for second order centered finite differences on non-uniform grids // Journal of Computational and Applied Mathematics. - 2017. - Vol.326. - P.1-14. - ISSN 0377-0427. - EISSN 1879-1778.
Внешние системы: DOI: 10.1016/j.cam.2017.05.006; РИНЦ: 31042963; SCOPUS: 2-s2.0-85019716220; WoS: 000405977300001;
Реферат: eng: In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon depending on the value of a free parameter. To this equation we apply an adaptive numerical method on redistributed grids. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. Moreover, we show also that some other choices of the nodes distributions lead to substantial degradation of the accuracy. This example is quite pedagogical and we use it only for illustrative purposes. It may serve as a guidance for more complex problems. © 2017 Elsevier B.V.
Ключевые слова: Supraconvergence; Second-order ordinary differential equations; Non-uniform grids; Finite differences; Complex problems; Central finite difference; Centered finite differences; Adaptive numerical methods; Numerical methods; Finite difference method; Differential equations; Boundary value problems; Supraconvergence; Non-uniform grids; Finite differences; Boundary value problems; Boundary layer; Ordinary differential equations; Boundary layers;
Издано: 2017
Физ. характеристика: с.1-14
Цитирование: 
1. Schlichting, H., Gersten, K., Boundary-Layer Theory, 2000, Springer-Verlag, Berlin, Heidelberg.
2. Roe, P.L., Computational fluid dynamics — retrospective and prospective. Int. J. Comput. Fluid Dyn. 19:8 (2005), 581–594.
3. Tikhonov, A.N., Samarskii, A.A., Homogeneous difference schemes. Zh. Vychisl. Mat. Mat. Fiz. 1:1 (1961), 5–63.
4. Tikhonov, A.N., Samarskii, A.A., Homogeneous difference schemes on non-uniform nets. Zh. Vychisl. Mat. Mat. Fiz. 2:5 (1962), 812–832.
5. Il'in, A.M., Differencing scheme for a differential equation with a small parameter affecting the highest derivative. Math. Notes Acad. Sci. USSR 6:2 (1969), 596–602.
6. Roos, H.-G., Ten ways to generate the Il'in and related schemes. J. Comput. Appl. Math. 53:1 (1994), 43–59.
7. Eymard, R., Fuhrmann, J., Gärtner, K., A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems. Numer. Math. 102:3 (2006), 463–495.
8. Numerov, B.V., Note on the numerical integration of d2xdt2=f(x,t). Astron. Nachr. 230:19 (1927), 359–364.
9. Numerov, B.V., A method of extrapolation of perturbations. Mon. Not. R. Astron. Soc. 84:8 (1924), 592–602.
10. Phaneendra, K., Rakmaiah, S., Reddy, M.C.K., Computational method for singularly perturbed boundary value problems with dual boundary layer. Procedia Eng. 127 (2015), 370–376.
11. Sudobicher, V.G., Shugrin, S.M., Flow along a dry channel. Izv. Akad. Nauk SSSR 13:3 (1968), 116–122.
12. Alalykin, G.B., Godunov, S.K., Kireyeva, L.L., Pliner, L.A., Solution of One-Dimensional Problems in Gas Dynamics on Moving Grids, 1970, Nauka, Moscow.
13. Farrell, P.A., Miller, J.J.H., O'Riordan, E., Shishkin, G.I., A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation. SIAM J. Numer. Anal. 33:3 (1996), 1135–1149.
14. Keller, H.B., Numerical methods in boundary-layer theory. Ann. Rev. Fluid Mech. 10:1 (1978), 417–433.
15. Ferreira, J.A., De Oliveira, P., Convergence properties of numerical discretizations and regridding methods. J. Comput. Appl. Math. 45:3 (1993), 321–330.
16. Samarskii, A.A., The Theory of Difference Schemes, 2001, CRC Press, New York.
17. Bakhvalov, N.S., The optimization of methods of solving boundary value problems with a boundary layer. USSR Comput. Math. Math. Phys. 9:4 (1969), 139–166.
18. Ferreira, J.A., Grigorieff, R.D., Supraconvergence and supercloseness of a scheme for elliptic equations on nonuniform grids. Numer. Funct. Anal. Optim. 27:5–6 (2006), 539–564.
19. Emmrich, E., Grigorieff, R.D., Supraconvergence of a finite difference scheme for elliptic boundary value problems of the third kind in fractional order Sobolev spaces. Comput. Methods Appl. Math. 6:2 (2006), 154–177.
20. Huang, W., Russell, R.D., Adaptive Moving Mesh Methods Applied Mathematical Sciences, vol. 174, 2011, Springer New York, New York, NY.
21. Budd, C., Dorodnitsyn, V., Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation. J. Phys. A: Math. Gen. 34:48 (2001), 10387–10400.
22. Budd, C.J., Piggott, M.D., The geometric integration of scale-invariant ordinary and partial differential equations. J. Comput. Appl. Math. 128:1–2 (2001), 399–422.
23. Chhay, M., Hoarau, E., Hamdouni, A., Sagaut, P., Comparison of some Lie-symmetry-based integrators. J. Comput. Phys. 230:5 (2011), 2174–2188.
24. Dorodnitsyn, V.A., Finite difference models entirely inheriting continuous symmetry of original differential equations. Internat. J. Modern Phys. C 05:04 (1994), 723–734.
25. Zel'dovich, Y.B., Raizer, Y.P., Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Dover ed., 2002, Dover Publications, New York.
26. Budd, C.J., Leimkuhler, B., Piggott, M.D., Scaling invariance and adaptivity. Appl. Numer. Math. 39:3–4 (2001), 261–288.
27. Barbeiro, S., Ferreira, J.A., Grigorieff, R.D., Supraconvergence of a finite difference scheme for solutions in Hs(0,L). IMA J. Numer. Anal. 25:4 (2005), 797–811.
28. Ferreira, J.A., Grigorieff, R.D., On the supraconvergence of elliptic finite difference schemes. Appl. Numer. Math. 28:2–4 (1998), 275–292.
29. Degtyarev, L.M., Drozdov, V.V., Ivanova, T.S., The method of adaptive grids for the solution of singularly perturbed one dimensional boundary value problems. Differ. Uravn. 23:7 (1987), 1160–1169.
30. Strang, G., Iserles, A., Barriers to stability. SIAM J. Numer. Anal. 20:6 (1983), 1251–1257.
31. Nikitin, A.G., On the principal eigenfunction of a singularly perturbed Sturm–Liouville problem. Zh. Vychisl. Mat. Mat. Fiz. 39:4 (1999), 588–591.
32. Godunov, S.K., Ryabenkii, V.S., Difference Schemes, 1987, North-Holland, Amsterdam.
33. Lax, P.D., Richtmyer, R.D., Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9:2 (1956), 267–293.
34. G.S. Khakimzyanov, D. Dutykh, D. E. Mitsotakis, N.Y. Shokina, Numerical solution of conservation laws on moving grids, J. Comput. Appl. Math. (2016) 1–28 (submitted for publication).
35. Fedorenko, R.P., Introduction to Computational Physics, 1994, MIPT Press, Moscow.
36. García-Archilla, B., Sanz-Serna, J.M., A finite difference formula for the discretization of d3dx3 on nonuniform grids. Math. Comp. 57:195 (1991), 239–257.
37. Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T., Kramer, P.B., Numerical Recipes: The Art of Scientific Computing, third ed., 2007, Cambridge University Press, Cambridge.