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Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation

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Abstract

A numerical method for solving systems of nonlinear one-dimensional balance laws, based on multivariate sigmoidal functions approximation, is developed. Constructive approximation theorems are first established in both, uniform and \(L^p\) norms. A priori as well as a posteriori error estimates are derived for the numerical solutions, when various kinds of sigmoidal functions, such as unit step, logistic and Gompertz functions, are chosen. The residual of the numerical method is also estimated. Numerical examples are given to test the performance of the algorithm, a comparison with the Godunov method is made concerning accuracy and computational cost. Finally, the numerical stability of the method is analyzed.

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Acknowledgments

The authors would like to thank the referees for their useful and constructive criticism, which lead to improve the quality of this paper. This work was supported, in part, by the GNAMPA and the GNFM of the Italian INdAM. Danilo Costarelli was partially supported by the Department of Mathematics and Computer Science, University of Perugia, and by the GNAMPA-INdAM Project “Metodi di approssimazione e applicazioni al Signal e Image Processing”, Project Number U2015/000396, 12/03/2015.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Perugia, 1, Via Vanvitelli, 06123, Perugia, Italy

    Danilo Costarelli

  2. Section of Mathematics, Department of Mathematics and Physics, University “Roma Tre”, 1, Largo S. Leonardo Murialdo, 00146, Rome, Italy

    Renato Spigler

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  1. Danilo Costarelli
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  2. Renato Spigler
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Correspondence to Danilo Costarelli.

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Communicated by Armin Iske.

To Peter D. Lax on his 90th birthday.

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Costarelli, D., Spigler, R. Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation. Comp. Appl. Math. 37, 99–133 (2018). https://doi.org/10.1007/s40314-016-0334-8

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  • DOI: https://doi.org/10.1007/s40314-016-0334-8

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