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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.

##### MSC:
 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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