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Second Order Implicit Schemes for Scalar Conservation Laws

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Numerical Mathematics and Advanced Applications ENUMATH 2015

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 112))

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Abstract

The today’s demands for simulation and optimization tools for water supply networks are permanently increasing. Practical computations of large water supply networks show that rather small time steps are needed to get sufficiently good approximation results – a typical disadvantage of low order methods. Having this application in mind we use higher order time discretizations to overcome this problem. Such discretizations can be achieved using so-called strong stability preserving Runge-Kutta methods which are especially designed for hyperbolic problems. We aim at approximating entropy solutions and are interested in weak solutions and variational formulations. Therefore our intention is to compare different space discretizations mostly based on variational formulations, and combine them with a second-order two-stage SDIRK method. In this paper, we will report on first numerical results considering scalar hyperbolic conservation laws.

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References

  1. S.E. Buckley, M.C. Leverett, Mechanism of fluid displacements in sands. Trans. AIME 146, 107–116 (1942)

    Article  Google Scholar 

  2. J.C. Butcher, Numerical Methods for Ordinary Differential Equations (John Wiley & Sons, England, 2003)

    Book  MATH  Google Scholar 

  3. B. Cockburn, An introduction to the discontinuous Galerkin method for convection-dominated problems, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics (Springer, Berlin/Heidelberg, 1997), pp. 151–268

    Google Scholar 

  4. D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69 (Springer, Heidelberg, 2012)

    Google Scholar 

  5. L.C. Evans, Partial Differential Equations (American Mathematical Society, Rhode Island, 2010)

    Book  MATH  Google Scholar 

  6. L. Ferracina, M.N. Spijker, Strong stability of singly-diagonally-implicit Runge-Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Gottlieb, C.W. Shu, Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Gottlieb, C.W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Gottlieb, D. Ketcheson, C.W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations (World Scientific, Singapore, 2011)

    Book  MATH  Google Scholar 

  10. W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33 (Springer, Heidelberg, 2003)

    Google Scholar 

  11. O. Kolb, J. Lang, Mathematical Optimization of Water Networks, Simulation and Continuous Optimization (Birkhäuser/Springer Basel AG, Basel, 2012)

    MATH  Google Scholar 

  12. O. Kolb, J. Lang, P. Bales, An implict box scheme for subsonic compressible flow with dissipative source term. Numer. Algorithms 53, 293–307 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. E.J. Kubatko, B.A. Yeager, D.I. Ketcheson, Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods. J. Sci. Comput. 60, 313–344 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.J. LeVeque, Numerical Methods for Conservation Laws. Lectures in Mathematics ETH Zürich (Birkhäuser Verlag, Basel/Boston/Berlin, 1990)

    Google Scholar 

  15. H. Martin, R. Pohl, Technische Hydromechanik 4 (Huss-Medien-GmbH, Berlin, 2000)

    Google Scholar 

  16. B. van Leer, Towards the ultimate conservation difference scheme. J. Comput. Phys. 32, 1–136 (1974)

    MATH  Google Scholar 

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Correspondence to Lisa Wagner .

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Wagner, L., Lang, J., Kolb, O. (2016). Second Order Implicit Schemes for Scalar Conservation Laws. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_4

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