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We present three-dimensional central finite volume methods for solving systems of hyperbolic equations. Based on the Lax-Friedrichs and Nessyahu-Tadmor one-dimensional central finite difference schemes, the numerical methods we propose involve an original and a staggered grid in order to avoid the resolution of the Riemann problems at the cell interfaces. The cells of the original grid are Cartesian (cubes with faces parallel to the axes) while those of the staggered grid are either Cartesian or diamond-shaped. We apply these methods and solve some ideal magnetohydrodynamics problems. To satisfy the solenoidal property of the magnetic field in the numerical solution, we present an adaptation of Evans and Hawley's constrained transport method for central schemes which we call ''CTCS''. The CTCS method is easy to implement, it deals directly with the computed solution and does not require any additional staggering for the magnetic field components; furthermore, it preserves the second-order accuracy of the base scheme. Even without the application of the CTCS procedure, our numerical base schemes do not break down, and may even in some cases deliver reasonable results. The diamond dual cell scheme has a slight advantage for shocks and contact discontinuities. Our numerical results are in good agreement with corresponding results appearing in the recent literature.

We present second-order accurate central finite volume methods adapted here to three-dimensional problems in ideal magnetohydrodynamics. These methods alternate between two staggered grids, thus leading to Riemann solver-free algorithms with relatively favorable computing times. The original grid considered in this paper is Cartesian, while the dual grid features either Cartesian or diamond-shaped oblique dual cells. 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 these methods and our results are in good agreement with existing results in the literature.

In this paper we propose a new well-balanced unstaggered central finite volume scheme for the shallow water equations on variable bottom topographies, with wet and dry states. Based on a special piecewise linear reconstruction of the cell-centered numerical solution and a careful discretization of the system of partial differential equations, the proposed numerical scheme ensures both a well-balanced discretization and the positivity requirement of the water height component. More precisely, the well-balanced requirement is fulfilled by following the surface gradient method, while the positivity requirement of the computed water height component is ensured by following a new technique specially designed for the unstaggered central schemes. The developed scheme is then validated and classical shallow water equation problems on variable bottom topographies with wet and dry states are successfully solved. The reported numerical results confirm the potential and efficiency of the proposed method

We propose a new well-balanced central finite volume scheme for the Ripa system both in one and two space dimensions. The Ripa system is a nonhomogeneous hyperbolic system with a non-zero source term that is obtained from the shallow water equations system by incorporating horizontal temperature gradients. The proposed numerical scheme is a second-order accurate finite volume method that evolves a non-oscillatory numerical solution on a single grid, avoids the process of solving Riemann problems arising at the cell interfaces, and follows a well-balanced discretization that ensures the steady state requirement by discretizing the geometrical source term according to the discretization of the flux terms. Furthermore the proposed scheme mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The proposed scheme is then applied and classical one and two-dimensional Ripa problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.

In this paper we present a new well‐balanced unstaggered central finite volume scheme for the numerical solution of the one‐dimensional Saint‐Venant systems on flat or variable bottom topographies. The Saint‐Venant equations also known as the shallow water equations (SWE) are useful when modeling the hydrodynamics of coastal oceans and lakes and the simulation of tsunami waves and inundation floods. The Saint‐Venant system is a hyperbolic system with a nonzero geometrical source term when flows over variable bottom topographies are considered. In this work we develop a new one‐dimensional well‐balanced central scheme for the numerical solution of SWE systems: The numerical base scheme is an unstaggered extension of the 1D Nessyahu and Tadmor central scheme that evolves the numerical solution on a unique grid and avoids the resolution of the Riemann problems arising at the cell interfaces thanks to a layer of ghost staggered cells. As is the case with general balance laws, the SWE system admits steady‐state solutions in which the nonzero divergence of the flux is exactly balanced by the source term. This balance is very difficult to maintain at the discrete level, and in general most numerical schemes for hyperbolic systems fail to preserve it, and thus generate nonphysical waves. In this work we propose a special discretization of the source term according to the discretization of the divergence of the flux by the numerical base scheme, and then adapt the surface gradient method to the case of one‐dimensional unstaggered central schemes. The resulting scheme ensures the well‐balanced constraint at the discrete level and is consistent with the SWE system.

A set of fully hyperbolic and fully conservative mixture conservation laws that treat the relative motion in gas-liquid two-phase flows are presented. These laws are based on compressible gas-liquid two-phase flow with velocity and pressure non-equilibrium. The resulting theoretical model accounts for the interaction between the gas and liquid phases with more reasonable physical meaning. Within this framework, the distinctness of the model conservation laws enables a straightforward extension and application of single-phase fluid dynamics numerical methods. Computations for a set of rarefaction waves test problem show that the interfacial friction source term has a strong influence on both the flow parameters and the wave structure in non-equilibrium two-phase flows.

An energy conserving scheme for the shallow water equations including the bathymetry term is derived. Energy conservation implies the well-balanced property. The soundness of the approach is verified by numerical examples. Namely the lake at rest and a dam break problem are presented. The new scheme is compared with an established scheme.

This paper reports on the application and development of a fully hyperbolic and fully conservative two‐phase flow model for the simulation of gas and magma flow within volcanic processes. The model solves a set of mixture conservation equations for the gas and magma two‐phase flow with velocity non‐equilibrium. In this model, the effect of the relative velocity is introduced by a kinetic constitutive equation with other equations for volume and mass fractions of the gas phase. The model is examined numerically by the widely used finite volume Godunov methods of centered‐type. Using the Riemann problem, we numerically simulate wave propagation and the development of shocks and rarefactions in volcanic eruptions. These simulations are of magma fragmentation type where the relative velocity continues to dominate. A series of test cases whose solution contains features relevant to gas–magma mixtures are conducted. In particular, numerical results indicate that the model implementation predicts key features of the relative velocity within volcanic processes without any mathematical or physical simplifications. Simulation results are sharply and accurately provided without any spurious oscillations in all of the flow variables. The numerical methods and results are also compared with other numerical methods available in the literature. It is found that the provided resolutions are more accurate for the considered test cases. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical simulations of natural convection heat and mass transfer in a square cavity using Comsol Multiphysics 5.0 software are presented. The effective thermal conductivity of nanofluid in porous media is computed using glass beads as porous media. It is observed that the heat and mass transfer rate increases with the increase of temperature variation as well as nanoparticle volume concentration.

Computations of gas-solid two-phase flow are carried out within the framework of mixture formulations. These computations are based on the Riemann problem for the resulting theoretical equations. The Riemann problem for the model equations is solved numerically using Godunov methods of centred-type. Numerical results are shown for carefully chosen test problems. The results show that both the model and the numerical methods provide very satisfactory solutions.

Two and three-dimensional finite volume extensions of the Lax-Friedrichs (LF) and Nessyahu-Tadmor (NT) one-dimensional difference schemes were previously presented and applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for inviscid or viscous compressible flows. These 'central' schemes by-pass the resolution of the Riemann problems, thanks to the use of the staggered Lax-Friedrichs scheme; two dual grids are used at alternate time steps. These methods are extended here to several problems in one- and multi-dimensional ideal compressible magnetohydrodynamics using a modified version of the first author's central methods with diamond shaped dual cells. In 2D the system has eight equations and solving the Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest that the accuracy is quite satisfactory. For our first numerical experiments, presented here, with one and two- dimensional problems, the divB=0 constraint has been satisfied by our scheme nearly up to the accuracy of the computations. For several more elaborate numerical tests in 2 and 3D, it turned out to be necessary to apply a specific strategy to enforce this constraint. The validity of our strategy, adapted from Evans-Hawley, is clearly confirmed by the results.

In this paper, we present a convergence analysis of a two-dimensional central finite volume scheme on unstructured triangular grids for hyperbolic systems of conservation laws. More precisely, we show that the solution obtained by the numerical base scheme presents, under an appropriate CFL condition, an optimal convergence to the unique entropy solution of the Cauchy problem.

We consider the Euler equations with gravitational source term and propose a new well-balanced unstaggered central finite volume scheme, which can preserve the hydrostatic balance state exactly. The proposed scheme evolves a nonoscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in time and space. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the gravitational source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the density and energy according to the discretization of steady state density and energy functions, respectively. Finally, several numerical experiments demonstrating the performance of the well-balanced schemes in both one and two spatial dimensions are presented. The results indicate that the new scheme is accurate, simple, and robust.

This paper presents further validation of previous works [1, 2] on simulating fully hyperbolic and fully conservative gas-solid mixture PDEs. Such a system allows non-equilibrium processes between the two phase systems and facilitates explosive volcanic eruption investigations. Resolutions for this system are presented and compared with other numerical methods demonstrating the fundamental physical and numerical significances to the relative motion within volcanic eruptions.

We propose a new one-dimensional unstaggered central scheme on nonuniform grids for the numerical solution of homogeneous hyperbolic systems of conservation laws with applications in two-phase flows and in hydrodynamics with and without gravitational effect. The numerical base scheme is a generalization of the original Lax–Friedrichs scheme and an extension of the Nessyahu and Tadmor central scheme to the case of nonuniform irregular grids. The main feature that characterizes the proposed scheme is its simplicity and versatility. In fact, the developed scheme evolves a piecewise linear numerical solution defined at the cell centers of a nonuniform grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of staggered cells used intermediately. Spurious oscillations are avoided using a slopes limiting procedure. The developed scheme is then validated and used to solve classical problems arising in gas–solid two phase flow problems. The proposed scheme is then extended to the case of non-homogenous hyperbolic systems with a source term, in particular to the case of Euler equations with a gravitational source term. The obtained numerical results are in perfect agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and potential of the proposed method to handle both homogeneous and non-homogeneous hyperbolic systems.

We propose two-dimensional central finite volume methods based on our multidimensional extensions of Nessyahu and Tadmor's one-dimensional non-oscillatory central scheme and a constrained transport-type method to solve ideal magnetohydrodynamic problems (MHD) and shallow water magnetohydrodynamic problems (SMHD). The main numerical scheme is second-order accurate both in space and time and uses an original Cartesian grid coupled to a Cartesian- or diamond-staggered dual grid to by-pass the resolution of the Riemann problems at the cell interfaces. To treat the non-vanishing magnetic field/flux divergence we have constructed an adaptation of Evans and Hawley's constrained transport method specifically designed for central schemes. Our numerical results show the efficiency and the potential of the scheme

Two and three-dimensional finite volume extensions of the Lax–Friedrichs (LF) and Nessyahu–Tadmor one-dimensional difference schemes were previously presented and successfully applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for both inviscid and viscous compressible flows. These “central” schemes by-pass the resolution, at the cell interfaces, of the Riemann problems, thanks to the use of the staggered LF scheme which serves as the base scheme on which high order finite volume methods can be constructed using van Leer’s MUSCL-type limited reconstruction principle. For this purpose, two dual grids are used at alternate time steps. These methods are extended here to several problems in one- and multi-dimensional ideal compressible magnetohydrodynamics using a modified version of the first author’s central methods with oblique (diamond shaped) dual cells. In two-dimensions the system has eight equations and solving the corresponding Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest the accuracy is quite satisfactory. In order to satisfy the physical constraint ∇ · B = 0, we have constructed a strategy (“CTCS”) inspired from the Constrained Transport method of Evans and Hawley. The validity of our base scheme and our CTCS approach is clearly confirmed by the results.

We propose an unstaggered, non-oscillatory, second-order accurate central scheme for approximating the solution of general hyperbolic systems in one and two space dimensions, and in particular for estimating the solution of ideal magnetohydrodynamic problems and shallow water magnetohydrodynamic problems. 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 unique grid, and avoids the resolution of the Riemann problems arising at the cell interfaces thanks to a layer of ghost staggered cells implicitly used while updating the numerical solution on the control cells. To satisfy the divergence-free constraint of the magnetic field/flux in the numerical solution of ideal/shallow water magnetohydrodynamic problems, we adapt Evans and Hawley's constrained transport method to our unstaggered base scheme and use it to correct the magnetic field/flux components at the end of each time step. The resulting method is used to solve classical ideal/shallow water magnetohydrodynamic problems; 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.

This paper presents high-resolution computations of a two-phase gas-solid mixture using a well-defined mathematical model. The HLL Riemann solver is applied to solve the Riemann problem for the model equations. This solution is then employed in the construction of upwind Godunov methods to solve the general initial-boundary value problem for the two-phase gas-solid mixture. Several representative test cases have been carried out and numerical solutions are provided in comparison with existing numerical results. To demonstrate the robustness, effectiveness and capability of these methods, the model results are compared with reference solutions. In addition to that, these results are compared with the results of other simulations carried out for the same set of test cases using other numerical methods available in the literature. The diverse comparisons demonstrate that both the model equations and the numerical methods are clear in mathematical and physical concepts for two-phase fluid flow problems.

We propose a new well-balanced unstaggered central finite volume scheme for hyperbolic balance laws with geometrical source terms. In particular we construct a new one and two-dimensional finite volume method for the numerical solution of shallow water equations on flat/variable bottom topographies. The proposed scheme evolves a non-oscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in space and time. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the geometrical source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the water height according to the discretization of the water level. The resulting scheme exactly satisfies the C-property at the discrete level. The proposed scheme is then applied and classical one and two-dimensional shallow water equation problems with flat or variable bottom topographies are successfully solved. The obtained numerical results are in good agreement with corresponding ones appearing in the recent literature, thus confirming the potential and efficiency of the proposed method.

We propose a new central finite volume scheme on unstructured triangular grids to approximate the solution of general two-dimensional hyperbolic systems of conservation laws. The proposed method is an unstructured two-dimensional extension of the original Nessyahu and Tadmor scheme, and a generalization of the barycentric central methods of Arminjon et al. Starting with a conformal finite element triangulation, the proposed method evolves a piecewise linear numerical solution on two staggered grids, thus avoiding the resolution of the Riemann problems arising at the cell interfaces. The control cells of the original grid are the triangles of a finite element mesh, while the dual cells are the staggered quadrilaterals constructed on adjacent triangles. The resulting central scheme is second-order accurate both in space and time and is oscillations-free thanks to numerical gradients limiting. The extension of the staggered Lax–Friedrichs scheme on unstructured grids is easily obtained from the Nessyahu and Tadmor extension by simply evolving a piecewise constant solution instead of a linear one. We validate the developed schemes and solve classical two-dimensional problems arising in gas dynamics. The quality of the obtained numerical results confirms the efficiency and robustness of the proposed schemes.