High-order IMEX-RK finite volume methods for multidimensional hyperbolic systems

  • E. Bertolazzi
  • G. Manzini
Conference paper


In this paper we present a high-order accurate cell-centered finite volume method for the semi-implicit discretization of multidimensional hyperbolic systems in conservative form on unstructured grids. This method is based on a special splitting of the physical flux function into convective and non-convective parts. The convective contribution to the global flux is treated implicitly by mimicking the upwinding of a scalar linear flux function while the rest of the flux is discretized in an explicit way. Spatial accuracy is ensured by allowing nonoscillatory polynomial reconstruction procedures, while time accuracy is attained by adopting a Runge-Kutta stepping scheme. The method can be considered naturally in the framework of the implicit-explicit (IMEX) schemes and the properties of the resulting operators are analysed using the properties of M-matrices.


Flux Function Numerical Flux Flux Form Convective Contribution Polynomial Reconstruction 
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  1. [1]
    Bertolazzi, E., Manzini, G. (2000): High-order IMEX-RK finite volume methods for multidimensional hyperbolic systems. Technical Report IAN-CNR-1202. Istituto di Analisi Numerica, Consiglio Nazionale delle Ricerche, PaviaGoogle Scholar
  2. [2]
    Davis, S.F. (1988): Simplified second-order Godunov-type methods. SIAM J. Sci. Statist. Comput. 9, 445–473MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Einfeldt, B. (1988): On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25, 294–318MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Einfeldt, B., Munz, C.-D., Roe, P.L., Sjögreen, B. (1991): On Godunov-type methods near low densities, J. Comput. Phys. 92, 273–295MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Godlewski, E., Raviart, P.-A. (1996): Numerical approximation of hyperbolic systems of conservation laws. Springer, BerlinMATHGoogle Scholar
  6. [6]
    Hirsch, C. (1990): Numerical computation of internal and external flows. Vol. 2. Computational methods for inviscid and viscous flows. Wiley, ChichesterGoogle Scholar
  7. [7]
    van Leer, B. (1979): Towards the ultimate conservative difference scheme. J. Comput. Phys. 32, 101–136CrossRefGoogle Scholar
  8. [8]
    van Leer, B. (1982): Flux-vector splitting for the Euler equations. In: Krause, E. (ed.): Numerical methods in fluid dynamics. (Lecture Notes in Physics, vol. 170). Springer, Berlin, pp. 507–512Google Scholar
  9. [9]
    Liotta, S.F., Romano, V., Russo, G. (2000): Central schemes for balance laws of relaxation type. SIAM J. Numer. Anal. 38, 1337–1356MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Liou, M.-S. (1996): A sequel to AUSM: AUSM+. J. Comput. Phys. 129, 364–382MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Pareschi, L., Russo, G. (2002): Implicit-explicit Runge-Kutta schemes for hyperbolic systems with stiff relaxation. UnpublishedGoogle Scholar
  12. [12]
    Pareschi, L., Russo, G. (2001): Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations. In: Trigante, D. (ed.): Recent trends in numerical analysis. Nova Science, New York, pp. 269–289Google Scholar
  13. [13]
    Shu, C.-W., Osher, S. (1989): Efficient implementation of essentially nonoscillatory shock-capturing schemes. II. J. Comput. Phys. 83, 32–78MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Steger, J.L., Warming, R.F. (1981): Flux-vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J. Comput. Phys. 40, 263–293MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Wada, Y., Liou, M.-S. (1997): An accurate and robust flux splitting scheme for shock and contact discontinuities. SIAM J. Sci. Comput. 18, 633–657MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Zeidler, E. (1986): Nonlinear functional analysis and its applications. I. Fixed-point theorems. Springer, New YorkMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • E. Bertolazzi
    • 1
  • G. Manzini
    • 2
  1. 1.Dipartimento di Ingegneria Meccanica e StrutturaleTrentoItaly
  2. 2.IMATI-CNRPaviaItaly

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