Abstract

Convergence is a major requirement for arbitrary approximate computation. For numerical study of convergence, in view of the definition of error, problems with analytical closed from solutions generally need to be at hand. In order to get rid of this necessity, in 2010, the concept of pseudo-error and pseudo-convergence are introduced for cases with geometrical changes of the algorithmic parameters. In 2014, the discussion is extended to arbitrary changes of the algorithmic parameters. In this study, it is displayed via examples and explained theoretically that for two solutions (obtained from two different computational methods) converging to the exact solution with similar rates, if in an interval of proper convergence the accuracy of the first solution is more/less than the second solution, the same can be claimed about the pseudo-errors, and vice versa. The implementations in practice are also discussed.

References

[1]    Henrici, P. (1962) Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, USA

[2]    Strikwerda, J. C. (1989) Finite Difference Schemes and Partial Differential Equations, Wadsworth & Books/Cole, USA.

[3]    Soroushian, A. (2010) Pseudo-convergence and its implementation in engineering approximate computations, The Proceedings of the 4th International Conference ‘From Scientific Computing to Computational Engineering’, 4th IC-SCCE, Athens, July.

[4]    Soroushian, A. (2014) Equivalence between convergence and pseudo-convergence when algorithmic parameters do not change geometrically, The Proceedings of the 6th International Conference ‘From Scientific Computing to Computational Engineering’,  6th IC-SCCE, Athens, July.