Stability study, error estimation, and condition number for semi-implicit schemes using incremental unknowns

✍ Scribed by Francois Pouit

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Volume: 12
• Category: Article
• ISSN: 0749-159X
• DOI: 10.1002/(sici)1098-2426(199611)12:6<743::aid-num7>3.0.co;2-s

✦ Synopsis

The concept of Incremental Unknowns leads to the introduction of large classes of new numerical schemes for which the large scale component I ' for the unknown, and its small scale component(s) Z are treated differently, leading to an improved condition number for elliptic problem and to an improved stability condition for evolution problems. In this article, we present a method for studying Y-explicit and Z-implicit schemes for the heat operator. For that purpose, we introduce a basis stemming from the eigenvectors of the Laplace operator, in which the amplification matrix is easy to analyze. This basis allows us to find the stability condition and the error scheme. We show rigorously that these schemes give a better stability condition, which becomes At 5 $ instead of At 5 for the classical explicit scheme in the case of two levels of unknowns. We show that it is possible to recover the initials errors of the schemes, if the constant v is small enough. Besides, the number of condition is better than that of the implicit scheme if At 2 ch' .