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 the finite difference approximation to the convection-diffusion equation. (English) Zbl 1100.65070
Summary: The system of ordinary differential equations arisen from discretizing the convection-diffusion equation with respect to the space variable to compute its approximate solution along a time level is considered. This system involves in computing $e^{k\bold A}y$ for some vector $y$, where $k$ is the time step-size and $\bold A$ is a large tridiagonal Toeplitz matrix. The common ways to compute an approximate solution of the convection-diffusion equation are based on replacing $e^{k\bold A}y$ by an its approximation. We give an explicit expression for the exact value of $e^{k\bold A}y$ and then the numerical results of our method with that of some well-known methods are compared.

65M06Finite difference methods (IVP of PDE)
65M20Method of lines (IVP of PDE)
35K15Second order parabolic equations, initial value problems
65F10Iterative methods for linear systems
Full Text: DOI
[1] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1996) · Zbl 0865.65009
[2] M. Gülsu, T. Öziş, Numerical solution of Burgers’ equation with restrictive Taylor approximation, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.01.106.
[3] Gallopoulos, E.; Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. stat. Comput. 13, 1236-1264 (1992) · Zbl 0757.65101
[4] Ismail, H. N. A.; Elbarbary, E. M. E.; Salem, G. S. E.: Restrictive Taylor’s approximation for solving convection -- diffusion equation. Appl. math. Comput. 147, 355-363 (2004) · Zbl 1034.65072
[5] Jain, M. K.: Numerical solution of differential equations. (1991) · Zbl 0744.76084
[6] D. Khojasteh Salkuyeh, Positive integer powers of the tridiagonal Toeplitz matrices, Int. Math. J., submitted for publication. · Zbl 1116.15022
[7] Meyer, C. D.: Matrix analysis and applied linear algebra. SIAM (2004)
[8] Moller, C.; Loan, C. V.: Nineteen dubious ways to compute the exponential of a matrix. SIAM rev. 20, 801-836 (1978) · Zbl 0395.65012
[9] Saad, Y.: Numerical solution of large Lyapunov equations. Proceedings of international symposium MTNS-89 3, 503-511 (1990) · Zbl 0719.65034
[10] Saad, Y.: Overview of Krylov subspace methods with applications to control problems. Proceedings of international symposium MTNS-89 3, 401-410 (1990) · Zbl 0722.93028
[11] Saad, Y.: Iterative methods for sparse linear systems. (1995)
[12] Smith, G. D.: Numerical solution of partial differential equations (finite difference methods). (1990)
[13] Strikwerda, J. C.: Finite difference schemes and partial differential equations. (1989) · Zbl 0681.65064
[14] Van Der Vorst, H. A.: An iterative solution method for solving $f(A)$x=b, using Krylov subspace information obtained for the symmetric positive definite matrix A. J. comput. Appl. math 18, 249-263 (1997) · Zbl 0621.65022
[15] Van Der Vorst, H. A.: Solution of $f(A)$x=b with projection methods for the matrix A. Lecture notes in computational science and engineering 15, 18-28 (2000) · Zbl 1187.65048