# 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”.

## References

- Aregba-Driollet D, Natalini R (2000) Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J Numer Anal 37(6):1973–2004MathSciNetzbMATHCrossRefGoogle Scholar
- Aregba-Driollet D, Briani M, Natalini R (2008) Asymptotic high-order schemes for $2 \times 2$ dissipative hyperbolic systems. SIAM J Numer Anal 46(2):869–894MathSciNetzbMATHCrossRefGoogle Scholar
- Bianchini S, Hanouzet B, Natalini R (2007) Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Commun Pure Appl Math 60:1559–1620MathSciNetzbMATHCrossRefGoogle Scholar
- Briani M, Natalini R (2006) Asymptotic high-order schemes for integro-differential problems arising in markets with jumps. Commun Math Sci 4(1):81–96MathSciNetzbMATHCrossRefGoogle Scholar
- Christoforou C, Trivisa K (2009) Sharp decay estimates for hyperbolic balance laws. J Differ Equ 247:401–423MathSciNetzbMATHCrossRefGoogle Scholar
- Costarelli D, Spigler R (2013) Solving Volterra integral equations of the second kind by sigmoidal functions approximation. J Integral Equ Appl 25(2):193–222MathSciNetzbMATHCrossRefGoogle Scholar
- Costarelli D, Spigler R (2014) A collocation method for solving nonlinear Volterra integro-differential equations of the neutral type by sigmoidal functions. J Integral Equ Appl 26(1):15–52MathSciNetzbMATHCrossRefGoogle Scholar
- Costarelli D, Spigler R (2018) Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation. Comput Appl Math 37(1):99–133MathSciNetzbMATHCrossRefGoogle Scholar
- Costarelli D, Laurenzi M, Spigler R (2013) Asymptotic-numerical solution of nonlinear systems of one-dimensional balance laws. J Comput Phys 245:347–363MathSciNetzbMATHCrossRefGoogle Scholar
- Dafermos CM, Hsiao L (1982) Hyperbolic systems of balance laws with inhomogeneity and dissipation. Indiana Univ Math J 31:471–491MathSciNetzbMATHCrossRefGoogle Scholar
- Hanouzet B, Natalini R (2003) Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch Ration Mech Anal 169:89–117MathSciNetzbMATHCrossRefGoogle Scholar
- Lax PD (1957) Hyperbolic systems of conservation laws II. Commun Pure Appl Math 10:537–566MathSciNetzbMATHCrossRefGoogle Scholar
- Lax PD (1972) The formation and decay of shock waves. Am Math Mon 79(3):227–241MathSciNetzbMATHCrossRefGoogle Scholar
- Lax PD (1973) Hyperbolic systems of conservation laws and the mathematical theory of shock waves. CBMS regional conf. series in appl. math., vol 11. Soc. Ind. Appl. Math., PhiladelphiaGoogle Scholar
- LeVeque RJ (2002) Finite volume methods for hyperbolic problems, Cambridge texts in applied mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Liu T-P (1979) Quasilinear hyperbolic systems. Commun Math Phys 68:141–172MathSciNetzbMATHCrossRefGoogle Scholar
- Natalini R, Ribot M (1996) An asymptotic high order mass-preserving scheme for a hyperbolic model of chemotaxis. SIAM J Numer Anal 33(1):1–16MathSciNetCrossRefGoogle Scholar
- Natalini R, Ribot M, Twarogowska M (2012) A well balanced numerical scheme for a one dimensional quasilinear hyperbolic model of chemotaxis. arXiv preprint arXiv:1211.4010 Google Scholar
- Roe PL (1987) Upwind differencing schemes for hyperbolic conservation laws with source terms. Nonlinear hyperbolic problems, (St. Etienne, 1986), Lecture notes in math., vol 1270. Springer, Berlin, pp 41–51Google Scholar
- Tadmor E (1984) The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme. Math Comput 43(168):353–368MathSciNetzbMATHCrossRefGoogle Scholar
- Tadmor E (2012) A review of numerical methods for nonlinear partial differential equations. Bull Am Math Soc (NS) 49(4):507–554MathSciNetzbMATHCrossRefGoogle Scholar
- Trèves F (1975) Basic linear partial differential equations, Pure appl. math. (Amst.), vol 62. Academic Press, New YorkGoogle Scholar