Numerical Algorithms

, Volume 33, Issue 1–4, pp 421–431

# Approximation and Prediction of the Numerical Solution of Some Burgers Problems

• Marc Prévost
• Denis Vekemans
Article

## Abstract

The Aitken's Δ2-prediction of Brezinski has already been used by Morandi Cecchi et al. in order to compute a numerical approximation of the solution of a parabolic initial-boundary value problem. This method consists in two consecutive steps: the first one is the approximation with a finite elements method, where the solution of the involved nonlinear system is computed by Gauss–Seidel method; the second one is a prediction of further terms with Aitken's Δ2-process. By comparison with this method, we use other methods of prediction in another way. First, we consider a generalization of Δ2-prediction, the so-called ε-prediction. In this paper, we only use vector prediction which is more stable than the scalar one. Then, the methods of prediction presented can be used in order to predict the starting vector of the Gauss–Seidel method.

approximation prediction ε-prediction vector prediction Burgers problem

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### References

1. [1]
C. Brezinski and M. Redivo-Zaglia, Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam, 1991).Google Scholar
2. [2]
M. Morandi Cecchi, R. Nociforo and P. Patuzzo Grego, Space-time finite elements numerical solution of Burgers problems, Le Matematiche LI (1996) 43–57.Google Scholar
3. [3]
M. Morandi Cecchi, R. Nociforo and P. Patuzzo Grego, Burgers problems, Theoretical results, Italian J. Pure Appl. Math. 2 (1997) 159–174.Google Scholar
4. [4]
M. Morandi Cecchi, M. Redivo-Zaglia and G. Scenna, Approximation of the numerical solution of parabolic problems, in: Computational and Applied Mathematics I, Algorithms and Theory, eds. C. Brezinski and U. Kulisch (North-Holland, Amsterdam, 1992) pp. 71–80.Google Scholar
5. [5]
M. Prévost and D. Vekemans, Partial Padé prediction, Numer. Algorithms 20 (1999) 23–50.Google Scholar
6. [6]
D. Priest, Algorithms for arbitrary precision floating point arithmetic, in: Proc. of the 10th Symposium on Computer Arithmetic, eds. P. Kornerup and D. Matula, Grenoble, France, 26–28 June 1991 (IEEE Computer Soc. Press, Los Alamitos, CA) pp. 132–145.Google Scholar
7. [7]
D. Vekemans, Algorithmes pour méthodes de prédiction, Thèse, Université des Sciences et Technologies de Lille (1995).Google Scholar
8. [8]
P. Wynn, On a device for computing the em(Sn) transformation, Math. Tables Aut. Comp. 10 (1956) 91–96.Google Scholar

## Authors and Affiliations

• Marc Prévost
• 1
• Denis Vekemans
• 1
1. 1.Laboratoire de Mathématiques Pures et AppliquéesUniversité du Littoral Côte d'Opale, Centre Universitaire de la Mi-VoixCalais CédexFrance