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A comparative study of Chebyshev acceleration and residue smoothing in the solution of nonlinear elliptic difference equations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1386)

Abstract

We compare the traditional and widely-used Chebyshev acceleration method with an acceleration technique based on residue smoothing. Both acceleration methods can be applied to a variety of function iteration methods and allow therefore a fair comparison. The effect of residue smoothing is that the spectral radius of the Jacobian matrix associated with the system of equations can be reduced substantially, so that the eigenvalues of the iteration matrix of the iteration method used are considerably decreased. Comparitive experiments clearly indicate that residue smoothing is superior to Chebyshev acceleration. For a model problem we show that the rate of convergence of the smoothed Jacobi process is comparable with that of ADI methods.

The smoothing matrices by which the residue smoothing is achieved, allow for a very efficient implementation, thus hardly increasing the computational effort of the iteration process. Another feature of residue smoothing is that it is directly applicable to nonlinear problems without affecting the algorithmic complexity. Moreover, the simplicity of the method offers excellent prospects for execution on vector and parallel computers.

The investigations were supported by the National Research Council (CNR) of Italy.

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References

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© 1989 Springer-Verlag

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Van der Houwen, P.J., Sommeijer, B.P., Pontrelli, G. (1989). A comparative study of Chebyshev acceleration and residue smoothing in the solution of nonlinear elliptic difference equations. In: Bellen, A., Gear, C.W., Russo, E. (eds) Numerical Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol 1386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089232

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  • DOI: https://doi.org/10.1007/BFb0089232

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51478-7

  • Online ISBN: 978-3-540-48144-7

  • eBook Packages: Springer Book Archive