Robust novel high‐order accurate numerical methods for singularly perturbed convection‐diffusion problems¹

This research was supported by the Russian Foundation for Basic Research under grants No. 04–01–00578, 04–01–89007‐NWO_a, by the Netherlands Research Organisation NWO under grant No. 047.016.008 and by the Boole Centre for Research in Informatics at the National University of Ireland, Cork.

Abstract

For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy.

Straipsnyje nagrinejami nedidelio tikslumo ϵ‐tolygiai konvertuojantys skaitmeniniai metodai, singuliariai sutrikdytiems kraštiniams uždaviniams. Paraboliniam konvekcijos‐difuzijos uždaviniui konvergavimo eile neviršija vienos antrosios, jeigu uždavinio duomenys yra pakankamai glodūs. Tačiau trūkiems pradiniams duomenims eile yra ne aukštesne už 2⁻¹. Šio tipo uždaviniams, naudojant naujai išvestus metodus, darbe sukonstruotos ϵ‐tolygiai konvertuojančios schemos aukštesniu tikslumu.

Key words:

• singular perturbation
• boundary layer
• parabolic convection‐diffusion equation
• difference scheme
• parameter‐uniform convergence
• high‐order accuracy

Notes

This research was supported by the Russian Foundation for Basic Research under grants No. 04–01–00578, 04–01–89007‐NWO_a, by the Netherlands Research Organisation NWO under grant No. 047.016.008 and by the Boole Centre for Research in Informatics at the National University of Ireland, Cork.