zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On a fourth-order finite-difference method for singularly perturbed boundary value problems. (English) Zbl 1160.65037
The authors presented a fourth-order finite difference method for singularly perturbed one-dimensional reaction-diffusion boundary-value problems. The domain is discretized by using a layer-adapted Bakhvalov-type mesh. Parameter-uniform convergence results are obtained. Numerical examples are provided to validate the theoretical fourth-order convergence of the method.

65L10Boundary value problems for ODE (numerical methods)
65L20Stability and convergence of numerical methods for ODE
65L12Finite difference methods for ODE (numerical methods)
34B15Nonlinear boundary value problems for ODE
34E15Asymptotic singular perturbations, general theory (ODE)
65L50Mesh generation and refinement (ODE)
Full Text: DOI
[1] Vulanović, R.: On a numerical solution of a type of singularly perturbed boundary value problem by using a special discretization mesh. Univ. novom sadu zb. Rad. prirod. Mat. fak. Ser. mat. 13, 187-201 (1983) · Zbl 0573.65064
[2] E. Bohl, Finite Modelle Gewöhnlicher Randwertaufgaben, Teubner, Stuttgart, 1981. · Zbl 0472.65070
[3] Miller, J. J. H.: Applications of advanced computational methods for boundary and interior layers. (1993) · Zbl 0782.00036
[4] Bakhvalov, N. S.: Towards optimization of methods for solving boundary value problems in the presence of a boundary layer. Zh. vychisl. Mat. i mat. Fiz. 9, 841-859 (1969)
[5] Boglaev, I. P.: Variacionno -- raznostnaya skhema dlya kraevoi zadachi s malym parametrom pri starshei proizvodnoi. Zh. vychisl. Mat. i mat. Fiz. 21, No. 4, 887-896 (1981)
[6] Clavero, C.; Lisbona, F.; Miller, J. J. H.: Uniform convergence of arbitrary order on nonuniform meshes for a singularly perturbed boundary value problem. J. comput. Appl. math. 59, 155-171 (1995) · Zbl 0856.65089
[7] Doolan, E. P.; Miller, J. J. H.; Schilders, W. H. A.: Uniform numerical methods for problems with initial and boundary layers. (1980) · Zbl 0459.65058
[8] Herceg, D.; Vulanović, R.: Some finite difference schemes for a singular perturbation problem on a nonuniform mesh. Univ. novom sadu zb. Rad. prirod. Mat. fak. Ser. mat. 11, 117-134 (1981) · Zbl 0506.65035
[9] Shishkin, G. I.: Raznostnaya skhema na neravnomernoi setke dlya differenci -- al’nogo uravneniya s malym parametrom pri starshei proizvodnoi. Zh. vychisl. Mat. i mat. Fiz. 23, No. 3, 609-619 (1983)
[10] R. Vulanović, Mesh Construction for Discretization of Singularly Perturbed Boundary Value Problems, Doctoral Dissertation, Faculty of Science, Univesity of Novi Sad, 1986.
[11] Herceg, D.: Uniform fourth order difference scheme for a singularly perturbation problem. Numer. math. 56, 675-693 (1990) · Zbl 0668.65060
[12] Clavero, C.; Gracia, J. L.: High order methods for elliptic and time dependent reaction -- diffusion singularly perturbed problems. Appl. math. Comput. 168, 1109-1127 (2005) · Zbl 1080.65062
[13] Bogucz, E. A.; Walker, J. D. A.: Fourth-order finite-difference methods for two-point boundary-value problems. IMA J. Numer. anal. 4, 69-82 (1984) · Zbl 0544.65052
[14] P. Deuflhard, G. Bader, Multiple shooting techniques revised, Preprint No. 163, University of Heidelberg, Inst. Für Angewandte Informatik, 1982.
[15] H.-J. Dieckhoff, P. Lory, H.J. Oberle, H.J. Pesch, P. Rentrop, R. Seidel, Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting, Report 7520, TU München, Inst. für Math, 1975. · Zbl 0361.65079
[16] Lentini, M.; Osborne, M. R.; Russel, R. D.: The close relationships between methods for solving two-point boundary value problems. SIAM J. Numer. anal. 22, 280-309 (1985) · Zbl 0575.65080
[17] Lentini, M.; Pereyra, V.: An adaptive finite difference solver for nonlinear two-point boundary problems with mild boundary layers. SIAM J. Numer. anal. 14, 91-111 (1977) · Zbl 0358.65069
[18] Stoer, J.; Bulirsch, R.: Einführung in die numerische Mathematik II. (1973) · Zbl 0257.65001
[19] Vulanović, R.; Herceg, D.; Petrović, N.: On the extrapolation for a singularly perturbed boundary value problem. Computing 36, 69-79 (1986) · Zbl 0576.34019
[20] Vulanović, R.: A priori meshes for singularly perturbed quasilinear two-point boundary value problems. IMA J. Numer. anal. 21, 349-366 (2001) · Zbl 0989.65081
[21] Varah, J. M.: A lower bound for the smallest singular value of a matrix. Linear algebra appl. 11, 3-5 (1975) · Zbl 0312.65028