Restoration of the contact surface in FORCE-type centred schemes I: Homogeneous two-dimensional shallow water equations

Advances in Water Resources

Volumn 47, Issue , 2012, Pages 88-99

  Canestrelli, Alberto ^a,b   Toro, Eleuterio F ^c  

^a UNIVERSITY OF PADOVA   (Italy)

^b BOSTON UNIVERSITY   (United States)

^c UNIVERSITY OF TRENTO   (Italy)

Author keywords

Centered schemes; Conservative hyperbolic systems; FORCE scheme; High order WENO finite volume methods; Shallow water equations; Unstructured mesh

Indexed keywords

CENTERED SCHEMES; FORCE SCHEME; HIGH-ORDER; HYPERBOLIC SYSTEM; SHALLOW WATER EQUATIONS; UNSTRUCTURED MESHES;

EULER EQUATIONS; FINITE VOLUME METHOD; MESH GENERATION; SHEAR WAVES;

MAGNETOHYDRODYNAMICS;

EULERIAN ANALYSIS; FINITE VOLUME METHOD; MEANDER; SHALLOW-WATER EQUATION; TWO-DIMENSIONAL MODELING;

EID: 84865758090     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2012.03.019     Document Type: Article

References (35)

1. Canestrelli A., Dumbser M., Siviglia A., 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 2010, 33(3):291-303.
2. Canestrelli A., Siviglia A., Dumbser M., 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 2009, 32(6):834-844.
3. Canestrelli A, Toro EF. Restoration of the contact surface in FORCE-type centred schemes II: Non-conservative one- and two-layer two-dimensional shallow water equations. Adv Water Resour 2012;this issue. http://dx.doi.org/10.1016/j.advwatres.2012.03.018. http://dx.doi.org/10.1016/j.advwatres.2012.03.018.
4. Castro M.J., Pardo A., Parés C., Toro E.F. On some fast well-balanced first order solvers for nonconservative systems. Math Comput 2010, 79:1427-1472.
5. Chow V.T. Open-channel hydraulics 1959, McGraw-Hill, New York.
6. Courant R., Isaacson E., Rees M. On the solution of nonlinear hyperbolic differential equations by finite differences. Comm Pure Appl Math 1952, 118:243-255.
7. de Saint-Venant A.J.C. Theorie du mouvement non-permanent des eaux, avec application aux crues des rivi'eres et 'a lintroduction des marees dans leur lit. CR Acad Sci Paris 1871, 73:147-154.
8. Dumbser M., Castro M., Hidalgo A., Parés C., Toro E.F. FORCE schemes on unstructured meshes. II: Non-conservative hyperbolic systems. Comput Methods Appl Mech Eng 2010, 199:625-647.
9. 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. J Comput Phys 2008, 227:8209-8253.
10. Dumbser M., Käser M. Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J Comput Phys 2007, 221:693-723.
11. Dumbser M., Käser M., Titarev V.A., Toro E.F. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007, 226:204-243.
12. Harten A. On a class of high resolution total-variation-stable finite-difference schemes. SIAM J Numer Anal 1984, 21:1-23.
13. Harten A., Lax P.D., Van Leer B. On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Rev 1983, 25:35-61.
14. Kurganov A., Lin C. On the reduction of numerical dissipation in central-upwind schemes. Commun Comput Phys 2007, 2(1):141-163.
15. Kurganov A., Petrova G. Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws. Numer Methods Partial Differ Eq 2005, 21(3):536-552.
16. Kurganov A., Tadmor E. New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J Comput Phys 2000, 160(1):241-282.
17. Lax P.D. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm Pure Appl Math 1954, 7(1):159-193.
18. Leopold L.B., Wolman M.G., Miller J.P. Fluvial processes in geomorphology 1964, W.H. Freeman and Co., 522p.
19. Levy D., Puppo G., Russo G. A fourth-order central weno scheme for multidimensional hyperbolic systems of conservation laws. SIAM J Sci Comput 2002, 24(2):480-506.
20. Li S. High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method. J Comput Phys 2008, 227(15):7368-7393.
21. Liu Y., Shu C.W., Tadmor E., Zhang M. Non-oscillatory hierarchical reconstruction for central and finite volume schemes. Commun Comput Phys 2007, 2(5):933-963.
22. Meister A., Struckmeier J. Hyperbolic partial differential equations 2002, Vieweg.
23. Nessyahu H., Tadmor E. Non-oscillatory central differencing for hyperbolic conservation-laws. J Comput Phys 1990, 87(2):408-463.
24. Pareschi L., Puppo G., Russo G. Central Runge-Kutta schemes for conservation laws. SIAM J Sci Comput 2005, 26(3):979-999.
25. Qiu J.X., Shu C.W. On the construction, comparison, and local characteristic decomposition for high-order central weno schemes. J Comput Phys 2002, 183(1):187-209.
26. Roe P.L. Approximate riemann solvers, parameter vectors, and difference-schemes. J Comput Phys 1981, 43(2):357-372.
27. Shu CW. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Technical Report 97-65, Langley Research Center, Hampton,Virginia, USA: NASA; 1997. p. 202-28.
28. Stecca G., Siviglia E.F., Toro A. Upwind-biased FORCE schemes with applications to free-surface shallow flows. J Comput Phys 2010, 229(18):6362-6380.
29. Stoker J.J. Water waves. The mathematical theory with applications 1992, JohnWiley and Sons.
30. E.F. Toro. On Glimm-related schemes for conservation laws. Technical Report MMU-9602. Manchester Metropolitan University, UK: Department of Mathematics and Physics; 1996.
31. Toro E.F., Spruce M., Speares W. Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 1994, 4:25-34.
32. Toro E.F. Shock capturing methods for free-surface shallow flows 2001, John Wiley and Sons, LDD.
33. Toro E.F. Riemann solvers and numerical methods for fluid dynamics 2009, Springer, Berlin. third ed.
34. Toro E.F., Hidalgo A., Dumbser M. FORCE schemes on unstructured meshes. I: Conservative hyperbolic systems. J Comput Phys 2009, 228:3368-3389.
35. Toro E.F., Titarev V.A. Solution of the generalized riemann problem for advection-reaction equations. Proc Roy Soc Lond Series A: Math Phys Eng Sci 2002, 458(2018):271-281.