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A family of finite volume Eulerian-Lagrangian methods for two-dimensional conservation laws. (English) Zbl 1315.65076
Summary: We develop a family of finite volume Eulerian-Lagrangian methods for the solution of nonlinear conservation laws in two space dimensions. The proposed approach belongs to the class of fractional-step procedures where the numerical fluxes are reconstructed using the modified method of characteristics, while an Eulerian method is used to discretize the conservation equation in a finite volume framework. The method is simple, accurate, conservative and it combines advantages of the modified method of characteristics to accurately solve the nonlinear conservation laws with an Eulerian finite volume method to discretize the equations. The proposed finite volume Eulerian-Lagrangian methods are conservative, non-oscillatory and suitable for hyperbolic or non-hyperbolic systems for which Riemann problems are difficult to solve or do not exist. Numerical results are presented for an advection-diffusion equation with known analytical solution. The performance of the methods is also analyzed on several applications in Burgers and Buckley-Leverett problems. The aim of such a method compared to the conventional finite volume methods is to solve nonlinear conservation laws efficiently and with an appropriate level of accuracy.

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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[1] LeVeque, R. J., (Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Bürich, (1992))
[2] Morton, K. W., Numerical solution of convection-diffusion problems, (1996), Chapman & Hall London · Zbl 0861.65070
[3] Raviart, P. A.; Godlewski, E., (Hyperbolic Systems of Conservation Laws, Collection Mathematiques et Applications, vol. 3-4, (1990), SMAI), Ellipses Eds · Zbl 1155.76374
[4] Ewing, R. E.; Wang, H., A summary of numerical methods for time-dependent advection-dominated partial differential equations, J. Comput. Appl. Math., 128, 423-445, (2001) · Zbl 0983.65098
[5] Russell, T. F.; Celia, M. A., An overview of research on eulerian-Lagrangian localized adjoint methods (ELLAM), Adv. Water Resour., 25, 1215-1231, (2002)
[6] Phillips, T. N.; Williams, A. J., A semi-Lagrangian finite volume method for Newtonian contraction flows, SIAM J. Sci. Comput., 22, 2152-2177, (2001) · Zbl 0996.76065
[7] Rui, H., A conservative characteristic finite volume element method for solution of the advection-diffusion equation, Comput. Methods Appl. Mech. Engrg., 197, 3862-3869, (2008) · Zbl 1194.76164
[8] Benkhaldoun, F.; Seaid, M., A simple finite volume method for the shallow water equations, J. Comput. Appl. Math., 234, 58-72, (2010) · Zbl 1273.76287
[9] Benkhaldoun, F.; Seaid, M., Combined characteristics and finite volume methods for sediment transport and bed morphology in surface water flows, Math. Comput. Simul., 81, 2073-2086, (2011) · Zbl 1419.76453
[10] Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys., 43, 357-372, (1981) · Zbl 0474.65066
[11] Douglas, J.; Russell, T. F., Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite elements or finite differences, SIAM J. Numer. Anal., 19, 871-885, (1982) · Zbl 0492.65051
[12] Robert, A., A stable numerical integration scheme for the primitive meteorological equations, Atmos. Ocean, 19, 35-46, (1981)
[13] Temperton, C.; Staniforth, A., An efficient two-time-level semi-Lagrangian semi-implicit integration scheme, Q. J. R. Meteorol. Soc., 113, 1025-1039, (1987)
[14] Bermejo, R., A Galerkin-characteristic algorithm for transport-diffusion equations, SIAM J. Numer. Anal., 32, 425-454, (1995) · Zbl 0854.65083
[15] Süli, E., Convergence and stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53, 459-483, (1988) · Zbl 0637.76024
[16] Pudykiewicz, J.; Staniforth, A., Some properties and comparative performance of the semi-Lagrangian method of robert in the solution of advection-diffusion equation, Atmos. Ocean., 22, 283-308, (1984)
[17] Christov, I.; Popov, B., New nonoscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws, J. Comput. Phys., 227, 5736-5757, (2008) · Zbl 1151.65068
[18] Liu, J.; Chen, H.; Ewing, R. E.; Qin, G., An efficient algorithm for characteristic tracking on two-dimensional triangular meshes, Computing, 80, 121-136, (2007) · Zbl 1120.65104
[19] Krisnamachari, S. V.; Hayes, L. J.; Russel, T. F., A finite element alternating-direction method combined with a modified method of characteristics for convection-diffusion problems, SIAM J. Numer. Anal., 26, 1462-1473, (1989) · Zbl 0693.65061
[20] Rusanov, V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR, 1, 267-279, (1961)
[21] Seaid, M., On the quasi-monotone modified method of characteristics for transport-diffusion problems with reactive sources, Comput. Methods Appl. Math., 2, 186-210, (2002) · Zbl 1075.76642
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