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Lectures on Hyperbolic Equations and Their Numerical Approximation

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Non-Newtonian Fluid Mechanics and Complex Flows

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2212))

Abstract

These introductory lecture notes on numerical methods for hyperbolic equations are suitable for advanced undergraduate and postgraduate students in mathematics and engineering disciplines. More advanced approaches exist and will be indicated as appropriate. The material is divided into four sections. Section 1 presents an overview of hyperbolic equations and also some basic concepts on numerical discretization techniques. Section 2 deals with a specific example, the system of non-linear shallow water equations; the equations are analysed and the Riemann problem is solved exactly in complete detail. In Sect. 3 we first present the Godunov method as applied to a generic hyperbolic system and then specialised to the shallow water system in one space dimension; approximate solution methods for the Riemann problem are also given. Finally, Sect. 4 gives a brief overview of the ADER approach to construct high-order numerical methods for hyperbolic equations, based on the first order Godunov method. Much of the material of these lectures has been taken from the author’s text books (Toro, Riemann solvers and numerical methods for fluid dynamics. A practical introduction, 3rd edn. Springer, Berlin (2009) and Toro, Shock-capturing methods for free-surface shallow flows. Wiley, Chichester (2001)), where further reading material can be found. I also recommend the textbook by Godlewski and Raviart (Numerical approximation of hyperbolic systems of conservation laws. Springer, New York (1996)) and that by LeVeque (Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge (2002)).

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References

  1. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 3rd edn. Springer, Berlin (2009). ISBN 978-3-540-25202-3. http://link.springer.com/book/10.1007%2Fb79761

    Book  Google Scholar 

  2. Toro, E.F.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, Chichester (2001)

    MATH  Google Scholar 

  3. Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York (1996)

    Book  Google Scholar 

  4. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

    Book  Google Scholar 

  5. Godunov, S.K.: A Finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Math. Sb. 47, 357–393 (1959)

    Google Scholar 

  6. Toro, E.F., Billett, S.J.: Centred TVD schemes for hyperbolic conservation laws. IMA J. Numer. Anal. 20, 47–79 (2000)

    Article  MathSciNet  Google Scholar 

  7. Harten, A., Lax, P.D., van Leer, B.: On spstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–36 (1983)

    Article  MathSciNet  Google Scholar 

  8. Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)

    Google Scholar 

  9. Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL Riemann solver. Technical Report, Department of Aerospace Science, College of Aeronautics, Cranfield Institute of Technology. CoA-9204 (1992)

    Google Scholar 

  10. Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL Riemann solver. Shock Waves 4, 25–34 (1994)

    Article  Google Scholar 

  11. Toro, E.F., Chakraborty, A.: Development of an approximate Riemann solver for the steady supersonic Euler equations. Aeronaut. J. 98, 325–339 (1994)

    Article  Google Scholar 

  12. Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic conservation laws. Math. Comput. 38(158), 339–374 (1982)

    Article  MathSciNet  Google Scholar 

  13. Dumbser, M., Toro, E.F.: A simple extension of the Osher Riemann solver to general non-conservative hyperbolic systems. J. Sci. Comput. 48, 70–88 (2011)

    Article  MathSciNet  Google Scholar 

  14. Dumbser, M., Toro, E.F.: On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Commun. Comput. Phys. 10, 635–671 (2011)

    Article  MathSciNet  Google Scholar 

  15. Toro, E.F.: Brain venous haemodynamics, neurological diseases and mathematical modelling. A review. Appl. Math. Comput. 272, 542–579 (2016)

    Article  MathSciNet  Google Scholar 

  16. Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

    Article  MathSciNet  Google Scholar 

  17. Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards very high-order Godunov schemes. In: Toro, E.F. (ed.) Godunov Methods: Theory and Applications. Edited Review. Conference in Honour of Godunov SK, vol. 1, pp. 897–902. Kluwer Academic/Plenum Publishers, New York (2001)

    Chapter  Google Scholar 

  18. Toro, E.F., Titarev, V.A.: Solution of the generalised Riemann problem for advection-reaction equations. Proc. R. Soc. London, Ser. A 458, 271–281 (2002)

    Google Scholar 

  19. Titarev, V.A., Toro, E.F.: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17, 609–618 (2002)

    Article  MathSciNet  Google Scholar 

  20. Dumbser, M., Balsara, D., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227, 8209–8253 (2008)

    Article  MathSciNet  Google Scholar 

  21. Dumbser, M., Schwartzkopff, T., Munz, C.D.: Arbitrary high order finite volume schemes for linear wave propagation. In: Computational Science and High Performance Computing II. Notes on Numerical Fluid Mechanics and Multidisciplinary Design Book Series (NNFM), vol. 91, pp. 129–144. Springer, Berlin (2006)

    Google Scholar 

  22. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  Google Scholar 

  23. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  Google Scholar 

  24. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  Google Scholar 

  25. Ben-Artzi, M., Falcovitz, J.: A second order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1–32 (1984)

    Article  MathSciNet  Google Scholar 

  26. Castro, C.E., Toro, E.F.: Solvers for the high-order Riemann problem for hyperbolic balance laws. J. Comput. Phys. 227, 2481–2513 (2008)

    Article  MathSciNet  Google Scholar 

  27. Dumbser, M., Enaux, C., Toro, E.F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971–4001 (2008)

    Article  MathSciNet  Google Scholar 

  28. Toro, E.F., Montecinos, G.I.: Implicit, semi-analytical solution of the generalized Riemann problem for stiff hyperbolic balance laws. J. Comput. Phys. 303, 146–172 (2015)

    Article  MathSciNet  Google Scholar 

  29. Götz, C.R., Iske, A.: Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws. Math. Comput. 85, 35–62 (2016)

    Article  MathSciNet  Google Scholar 

  30. Götz, C.R., Dumbser, M.: A novel solver for the generalized Riemann problem based on a simplified LeFloch-Raviart expansion and a local space-time discontinuous Galerkin formulation. J. Sci. Comput. 69(2), 805–840 (2016)

    Article  MathSciNet  Google Scholar 

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Correspondence to Eleuterio F. Toro .

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Toro, E.F. (2018). Lectures on Hyperbolic Equations and Their Numerical Approximation. In: Farina, A., Mikelić, A., Rosso, F. (eds) Non-Newtonian Fluid Mechanics and Complex Flows. Lecture Notes in Mathematics(), vol 2212. Springer, Cham. https://doi.org/10.1007/978-3-319-74796-5_3

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