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)).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
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
Toro, E.F.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, Chichester (2001)
Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer, New York (1996)
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)
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)
Toro, E.F., Billett, S.J.: Centred TVD schemes for hyperbolic conservation laws. IMA J. Numer. Anal. 20, 47–79 (2000)
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)
Rusanov, V.V.: Calculation of interaction of non-steady shock waves with obstacles. J. Comput. Math. Phys. USSR 1, 267–279 (1961)
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)
Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL Riemann solver. Shock Waves 4, 25–34 (1994)
Toro, E.F., Chakraborty, A.: Development of an approximate Riemann solver for the steady supersonic Euler equations. Aeronaut. J. 98, 325–339 (1994)
Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic conservation laws. Math. Comput. 38(158), 339–374 (1982)
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)
Dumbser, M., Toro, E.F.: On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Commun. Comput. Phys. 10, 635–671 (2011)
Toro, E.F.: Brain venous haemodynamics, neurological diseases and mathematical modelling. A review. Appl. Math. Comput. 272, 542–579 (2016)
Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981)
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)
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)
Titarev, V.A., Toro, E.F.: ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17, 609–618 (2002)
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)
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)
Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)
Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Ben-Artzi, M., Falcovitz, J.: A second order Godunov-type scheme for compressible fluid dynamics. J. Comput. Phys. 55, 1–32 (1984)
Castro, C.E., Toro, E.F.: Solvers for the high-order Riemann problem for hyperbolic balance laws. J. Comput. Phys. 227, 2481–2513 (2008)
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)
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)
Götz, C.R., Iske, A.: Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws. Math. Comput. 85, 35–62 (2016)
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-74796-5_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-74795-8
Online ISBN: 978-3-319-74796-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)