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
S.E. Buckley, M.C. Leverett, Mechanism of fluid displacements in sands. Trans. AIME 146, 107–116 (1942)
J.C. Butcher, Numerical Methods for Ordinary Differential Equations (John Wiley & Sons, England, 2003)
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
D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications, vol. 69 (Springer, Heidelberg, 2012)
L.C. Evans, Partial Differential Equations (American Mathematical Society, Rhode Island, 2010)
L. Ferracina, M.N. Spijker, Strong stability of singly-diagonally-implicit Runge-Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)
S. Gottlieb, C.W. Shu, Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998)
S. Gottlieb, C.W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
S. Gottlieb, D. Ketcheson, C.W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations (World Scientific, Singapore, 2011)
W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33 (Springer, Heidelberg, 2003)
O. Kolb, J. Lang, Mathematical Optimization of Water Networks, Simulation and Continuous Optimization (Birkhäuser/Springer Basel AG, Basel, 2012)
O. Kolb, J. Lang, P. Bales, An implict box scheme for subsonic compressible flow with dissipative source term. Numer. Algorithms 53, 293–307 (2010)
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)
R.J. LeVeque, Numerical Methods for Conservation Laws. Lectures in Mathematics ETH Zürich (Birkhäuser Verlag, Basel/Boston/Berlin, 1990)
H. Martin, R. Pohl, Technische Hydromechanik 4 (Huss-Medien-GmbH, Berlin, 2000)
B. van Leer, Towards the ultimate conservation difference scheme. J. Comput. Phys. 32, 1–136 (1974)
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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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DOI: https://doi.org/10.1007/978-3-319-39929-4_4
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