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A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

Published online by Cambridge University Press:  20 August 2015

Guglielmo Stecca ,
Annunziato Siviglia  and
Eleuterio F. Toro
Guglielmo Stecca*
Affiliation:
Department of Civil and Environmental Engineering, University of Trento, Via Mesiano 77, I-38100 Trento, Italy
Annunziato Siviglia*
Affiliation:
Department of Civil and Environmental Engineering, University of Trento, Via Mesiano 77, I-38100 Trento, Italy
Eleuterio F. Toro*
Affiliation:
Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, I-38100 Trento, Italy
*
Corresponding author.Email:guglielmo.stecca@ing.unitn.it
Email:nunzio.siviglia@ing.unitn.it
Email:toroe@ing.unitn.it
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Abstract

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro, in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2 x 2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.

Keywords

65M0876M12Conservative hyperbolic systemscentred schemesunstructured meshesnumerical fluxesshallow water equationsFORCEupwind-biased
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 1183 - 1214
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Arminjon, P. and St-Cyr, A., Nessyahu-Tadmor-type central finite volume methods without predictor for 3D Cartesian and unstructured tetrahedral grids, Appl. Numer. Math., 46(2):135–155, 2003.Google Scholar
[2]Canestrelli, A., Dumbser, M., Siviglia, A. and Toro, E. F., Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed, Adv. Water Resour., 33(3):291–303, 2010.Google Scholar
[3]Canestrelli, A., Siviglia, A., Dumbser, M. and Toro, E. F., Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed, Adv. Water Resour., 32(6):834–844, 2009.Google Scholar
[4]Casper, J., Atkins, H. L., A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems, J. Comput. Phys., 106:62–76, 1993.Google Scholar
[5]Dumbser, M., Enaux, C. and Toro, E. F., Finite volume schemes of very high order for stiff hyperbolic balance laws, J. Comput. Phys., 227(8):3971–4001, 2008.Google Scholar
[6]Dumbser, M., Hidalgo, A., Castro, M., Pares, C. and Toro, E. F., FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems, Comput. Method. Appl. M., 199(9– 12):625–647, 2010.Google Scholar
[7]Dumbser, M. and Käser, M., Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221:693–723, 2007.Google Scholar
[8]Dumbser, M., Käser, M., Titarev, V. A. and Toro, E. F., Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems, J. Comput. Phys., 226:204–243, 2007.Google Scholar
[9]Godunov, S. K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47:271–306, 1959.Google Scholar
[10]Harten, A., Engquist, B., Osher, S. and Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71:231–303, 1987.Google Scholar
[11]Harten, A., Lax, P. D. and van Leer, B., On upstream differencing and Godunov-type schemes, SIAM Rev., 25(1):35–61, 1983.Google Scholar
[12]Jiang, G. S. and Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126:202–228, 1996.CrossRefGoogle Scholar
[13]Jiang, G. S. and Tadmor, E., Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19(6):1892–917,1998.Google Scholar
[14]Kurganov, A., Noelle, S. and Petrova, G., Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23(3):707–740, 2001.Google Scholar
[15]Kurganov, A. and Petrova, G., Central schemes and contact discontinuities, ESAIM-Math. Model. Num., 34(6):1259–1275, 2000.Google Scholar
[16]Kurganov, A. and Petrova, G., Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws, Numer. Meth. Part. D. E., 21(3):536–552, 2005.CrossRefGoogle Scholar
[17]Kurganov, A. and Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160(1):241–282, 2000.Google Scholar
[18]Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure. Appl. Math., VII:159–193, 1954.Google Scholar
[19]Liu, X. D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115:200–212, 1994.Google Scholar
[20]Munz, C. D., On the numerical dissipation of high resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 77:18–39, 1998.Google Scholar
[21]Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation-laws, J. Comput. Phys., 87(2):408–463, 1990.Google Scholar
[22]Ricchiuto, M. and Bollermann, A., Stabilized residual distribution for shallow water simulations, J. Comput. Phys., 228(4):1071–1115, 2009.Google Scholar
[23]Roe, P. L., Some contributions to the modelling of discontinuous flows, in: Proceedings of the SIAM/AMS Seminar, 1983.Google Scholar
[24]Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77:439–471, 1988.Google Scholar
[25]Stecca, G., Siviglia, A. and Toro, E. F., Upwind-biased FORCE schemes with applications to free-surface shallow flows, J. Comput. Phys., 229(18):6362–6380, 2010.Google Scholar
[26]Toro, E. F., Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley and Sons Ltd, 2001.Google Scholar
[27]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Third Edition. Springer-Verlag, 2009.Google Scholar
[28]Toro, E. F. and Billett, S. J., Centred TVD schemes for hyperbolic conservation laws, IMA J. Numer. Anal., 20:47–79, 2000.Google Scholar
[29]Toro, E. F., Hidalgo, A. and Dumbser, M., FORCE schemes on unstructured meshes I: Conservative hyperbolic systems, J. Comput. Phys., 228(9):3368–3389, 2009.CrossRefGoogle Scholar
[30]Toro, E. F., Millington, R. C. and Nejad, L. A. M., Towards very high order Godunov schemes, in: Toro, E. F. (Ed.), Godunov Methods: Theory and Applications, Kluwer/Plenum Academic Publishers, 907–940, 2001.Google Scholar
[31]Toro, E. F. and Siviglia, A., PRICE: primitive centred schemes for hyperbolic systems, Int. J. Numer. Methods Fluids, 42(12):1263–1291, 2003.Google Scholar
[32]Toro, E. F. and Titarev, V. A., Solution of the generalized Riemann problem for advection-reaction equations, Proc. R. Soc. A-Math. Phys. Eng. Sci., 458:271–281, 2002.Google Scholar
[33]van Leer, B., Towards the ultimate conservative difference scheme II: Monotonicity and conservation combined in a second order scheme, J. Comput. Phys., 14:361–370, 1974.Google Scholar