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



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.


Systems of balance laws Dissipative balance laws Asymptotic-numerical methods 

Mathematics Subject Classification

41A30 65M25 35L04 35L65 



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”.


  1. Aregba-Driollet D, Natalini R (2000) Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J Numer Anal 37(6):1973–2004MathSciNetCrossRefMATHGoogle Scholar
  2. 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):869–894MathSciNetCrossRefMATHGoogle Scholar
  3. 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 60:1559–1620MathSciNetCrossRefMATHGoogle Scholar
  4. Briani M, Natalini R (2006) Asymptotic high-order schemes for integro-differential problems arising in markets with jumps. Commun Math Sci 4(1):81–96MathSciNetCrossRefMATHGoogle Scholar
  5. Christoforou C, Trivisa K (2009) Sharp decay estimates for hyperbolic balance laws. J Differ Equ 247:401–423MathSciNetCrossRefMATHGoogle Scholar
  6. Costarelli D, Spigler R (2013) Solving Volterra integral equations of the second kind by sigmoidal functions approximation. J Integral Equ Appl 25(2):193–222MathSciNetCrossRefMATHGoogle Scholar
  7. 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):15–52MathSciNetCrossRefMATHGoogle Scholar
  8. Costarelli D, Spigler R (2018) Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation. Comput Appl Math 37(1):99–133MathSciNetCrossRefGoogle Scholar
  9. Costarelli D, Laurenzi M, Spigler R (2013) Asymptotic-numerical solution of nonlinear systems of one-dimensional balance laws. J Comput Phys 245:347–363MathSciNetCrossRefMATHGoogle Scholar
  10. Dafermos CM, Hsiao L (1982) Hyperbolic systems of balance laws with inhomogeneity and dissipation. Indiana Univ Math J 31:471–491MathSciNetCrossRefMATHGoogle Scholar
  11. Hanouzet B, Natalini R (2003) Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch Ration Mech Anal 169:89–117MathSciNetCrossRefMATHGoogle Scholar
  12. Lax PD (1957) Hyperbolic systems of conservation laws II. Commun Pure Appl Math 10:537–566MathSciNetCrossRefMATHGoogle Scholar
  13. Lax PD (1972) The formation and decay of shock waves. Am Math Mon 79(3):227–241MathSciNetCrossRefMATHGoogle Scholar
  14. 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
  15. LeVeque RJ (2002) Finite volume methods for hyperbolic problems, Cambridge texts in applied mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  16. Liu T-P (1979) Quasilinear hyperbolic systems. Commun Math Phys 68:141–172MathSciNetCrossRefMATHGoogle Scholar
  17. 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
  18. 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
  19. 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
  20. Tadmor E (1984) The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme. Math Comput 43(168):353–368MathSciNetCrossRefMATHGoogle Scholar
  21. Tadmor E (2012) A review of numerical methods for nonlinear partial differential equations. Bull Am Math Soc (NS) 49(4):507–554MathSciNetCrossRefMATHGoogle Scholar
  22. Trèves F (1975) Basic linear partial differential equations, Pure appl. math. (Amst.), vol 62. Academic Press, New YorkGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer SciencesUniversity of PerugiaPerugiaItaly
  2. 2.Department of Mathematics and PhysicsRoma Tre UniversityRomeItaly

Personalised recommendations