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In this paper we develop a new central unstaggered finite volume method for solving systems of hyperbolic equations. Based on the Lax‐Friedrichs central scheme and on the Nessyahu and Tadmor (NT) one‐dimensional non‐oscillatory central scheme, we construct a new class of unstaggered second‐order non‐oscillatory central finite volume schemes for approximating solutions of hyperbolic systems of conservation laws. In contrast with the original (NT) central scheme that evolves the numerical solution on an original grid (at even time steps) and on a staggered one (at odd time steps), the method we propose evolves the numerical solution on a single grid and uses a “ghost” staggered grid to avoid the time consuming resolution of the Riemann problems arising at the cell interfaces. The numerical solution is defined on the computational domain using piecewise linear interpolants. To avoid undesired oscillations a slope limiting process is applied; this results in a scheme that is second‐order accurate in space. To guarantee second‐order accuracy in time a second‐order quadrature rule is applied. We apply our numerical scheme and solve some classical shallow water equation problems. The numerical results presented in this work show the efficiency and the potential of our unstaggered central scheme; they compare very well with those obtained using the original (NT) central scheme and are in a very good agreement with corresponding results appearing in the recent literature.

This contribution report on recent scientific methodologies concerning compressible gas-liquid two-phase mixture. Within this context, a one-dimensional set of three partial differential equations that takes account of the velocity nonequilibrium between the two phases is used. Motivated by the hyperbolicity and conservativity features of these equations, a closed-form theoretical solution has provided key insights into the Riemann problem for the gas-liquid two-phase mixture. In addition to that, the proposed theoretical solution, that is an exact Riemann solver, calculates the entire wave structure including the middle states of the mixture and phases of the flow. To this end, the exact Riemann solver guarantees that the system of partial differential equations is independent of the type of the numerical technique used to validate it. Consequently, the model equations are solved numerically using Godunov methods of the upwind-type based upon the proposed exact Riemann solver. Sample academic simulations are performed and validated against analytical observations for the gas-liquid two-phase mixture. Through these simulations, we compare the predicted results with other numerical methods such as the Riemann-free solvers. Excellent agreement is observed between the analytical and numerical simulations.

We present second‐order accurate central finite volume methods adapted to three‐dimensional ideal magnetohydrodynamics problems. These methods alternate between two staggered grids, thus leading to Riemann solver‐free algorithms with relatively favorable computing times. The div·B = 0 constraint on the magnetic field is enforced with a suitable adaptation of the constrained transport method to our central schemes. Numerical experiments show the feasibility of the proposed methods and our results are in good agreement with existing results in the recent literature. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

In this paper we present a two-dimensional extension of the staggered Lax-Friedrichs scheme for the approximate solution of hyperbolic conservation laws on unstructured grids. By evolving the numerical solution on two staggered grids, the proposed scheme avoids the resolution of the Riemann problems arising at the cell interfaces. The control cells of the original grid are regular triangular cells of a finite element triangulation, while the staggered dual cells are quadrilaterals constructed on the triangular cells of the original grid. The accuracy and stability of the scheme are investigated and classical problems arising in gas dynamics are solved. The obtained results are in good agreement with corresponding ones appearing in the literature thus confirming the efficiency and potential of the proposed method.

We develop an unstaggered central scheme for approximating the solution of general two‐dimensional hyperbolic systems. In particular, we are interested in solving applied problems arising in hydrodynamics and astrophysics. In contrast with standard central schemes that evolve the numerical solution on two staggered grids at consecutive time steps, the method we propose evolves the numerical solution on a single grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of ghost cells implicitly used. The numerical base scheme is used to solve shallow water equation problems and ideal magnetohydrodynamic problems. To satisfy the divergence‐free constraint of the magnetic field in the numerical solution of ideal magnetohydrodynamic problems, we adapt Evans and Hawley’s the constrained transport method to our unstaggered base scheme, and apply it to correct the magnetic field components at the end of each time step. The obtained results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and the potential of the proposed method.

In this paper we extend a central finite volume method on nonuniform grids to the case of drift-flux two-phase flow problems. The numerical base scheme is an unstaggered, non oscillatory, second-order accurate finite volume scheme that evolves a piecewise linear numerical solution on a single grid and uses dual cells intermediately while updating the numerical solution to avoid the resolution of the Riemann problems arising at the cell interfaces. We then apply the numerical scheme and solve a classical drift-flux problem. The obtained results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential of the proposed scheme.