# An efficient asymptotic-numerical method to solve nonlinear systems of one-dimensional balance laws

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## Abstract

An asymptotic-numerical method to solve the initial-boundary value problems for systems of balance laws in one space dimension, on the half space is developed. Expansions in powers of \(t^{-1/2}\) are used, in view of the precise asymptotic behavior recently established on theoretical bases. This approach increases considerably the efficiency of a previous one, where just expansions in inverse powers of *t* were made. Numerical examples and comparisons with the Godunov, the asymptotic high order, and the asymptotic-numerical method earlier developed are presented. Expanding the solution in powers of \(t^{-1/2}\) instead of \(t^{-1}\), a saving of about one-half of the CPU time can be realized, still achieving the same accuracy.

## Keywords

Systems of balance laws Dissipative balance laws Asymptotic-numerical methods## Mathematics Subject Classification

41A30 65M25 35L04 35L65## Notes

### Acknowledgements

This work was carried out within the framework of the research groups GNAMPA and GNFM of the Italian INdAM. DC was partially supported by the Department of Mathematics and Computer Sciences, University of Perugia (Italy), and by the 2017 GNAMPA-INdAM Project “Approssimazione con operatori discreti e problemi di minimo per funzionali del calcolo delle variazioni con applicazioni all’imaging”.

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