A uniformly convergent alternating direction HODIE finite difference scheme for 2D time-dependent convection-diffusion problems

IMA Journal of Numerical Analysis

Volumn 26, Issue 1 SPEC. ISS., 2006, Pages 155-172

  Clavero, C ^a   Gracia, J L ^a   Jorge, J C ^b  

^a UNIVERSITY OF ZARAGOZA   (Spain)

^b PUBLIC UNIVERSITY OF NAVARRE   (Spain)

Author keywords

Alternating directions; HODIE schemes; Shishkin meshes; Singular perturbation; Uniform convergence

Indexed keywords

DIFFUSION IN LIQUIDS; HEAT CONVECTION; NUMERICAL METHODS;

ALTERNATING DIRECTIONS; HODIE SCHEMES; SHISHKIN MESHES; SINGULAR PERTURBATIONS; UNIFORM CONVERGENCE;

FINITE DIFFERENCE METHOD;

EID: 30344482243     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/dri029     Document Type: Article

References (18)

1. BRENNER, P. & THOMÉE, V. (1979) On rational approximations of semigroups. SIAM J. Numer. Anal., 16, 683-694.
2. BUJANDA, B. & JORGE, J. C. (1999) Third order fractional step methods for multidimensional evolutionary convection-diffusion problems. Finite Difference Methods, Theory and Applications (L. G. Vulkov, J. J. H. Miller & G. I. Shishkin, eds). Nova Science, pp. 110-117.
3. CLAVERO, C., GRACIA, J. L. & LISBONA, F. (1999) High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. Numer. Algorithms, 22, 73-97.
4. CLAVERO, C., JORGE, J. C., LISBONA, F. & SHISHKIN, G. I. (1998) A fractional step method on a special mesh for the resolution of multidimensional evolutionary convection-diffusion prolems. Appl. Numer. Math., 27, 211-231.
5. FARRELL, P., HEGARTY, A., MILLER, J. J. H., O'RIORDAN, E. & SHISHKIN, G. I. (2000) Robust Computational Techniques for Boundary Layers. Boca Raton: Chapman and Hall/CRC Press.
6. FRÖHNER, A., LINSS, T. & ROOS, H. G. (2001) Defect correction on Shishkin-type meshes. Numer. Algorithms, 26, 281-299.
7. GRACIA, J. L., JORGE, J. C., BUJANDA, B. & CLAVERO, C. (2003) High order uniformly convergent fractional step RK methods and HODIE finite difference schemes for 2D evolutionary convection-diffusion problems. J. Comput. Methods Sci. Eng., 3, 403-413.
8. HEMKER, P. W., SHISHKIN, G. I. & SHISHKINA, L. P. (2000) ε-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems. IMA J. Numer. Anal., 20, 99-121.
9. JORGE, J. C. & BUJANDA, B. (2000) High order fractionary step schemes for evolutionary convection-diffusion-reaction problems. Problems in Modern Applied Mathematics, pp. 31-37.
10. KELLOGG, R. B. & SHIH, S. (1987) Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal., 18, 1467-1511.
11. KOPTEVA, N. V. (1997) On the uniform in small parameter convergence of a weighted scheme for the one-dimensional time-dependent convection-diffusion equation. Comput. Math. Math. Phys., 17, 1173-1180.
12. LADYZHENSKAJA, O. A., SOLONNIKOV, V. A. & URAL'TSEVA, N. N. (1968) Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, vol. 23. Providence, RI: American Mathematical Society.
13. LINSS, T. (1999) An upwind difference scheme on a novel Shishkin-type mesh for a linear convection-diffusion problem. J. Comput. Appl. Math., 110, 93-104.
14. LINSS, T. & STYNES, M. (1999) A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problem. Appl. Numer. Math., 31, 255-270.
15. MATHEW, T. P., POLYAKOV, P. L., RUSSO, G. & WANG, J. (1998) Domain decomposition operator splittings for the solution of parabolic equations. SIAM J. Sci. Comput., 19, 912-932.
16. PALENCIA, C. (1993) A stability result for sectorial operators in Banach spaces. SIAM J. Numer. Anal., 30, 1373-1384.
17. PEACEMAN, D. & RACHFORD, H. (1955) The numerical solution of elliptic and parabolic differential equations. J. SIAM, 3, 28-41.
18. SHISHKIN, G. I. (1990) Grid approximation of singularly perturbed boundary value problems with convective terms. Sov. J. Numer. Anal. Math. Model., 5, 173-187.