R. Touma

Central unstaggered finite volume schemes for hyperbolic systems: Applications to unsteady shallow water equations

A class of central unstaggered finite volume methods for approximating solutions of hyperbolic systems of conservation laws is developed in this paper. The proposed method is an extension of the central, non-oscillatory, finite volume method of Nessyahu and Tadmor (NT). In contrast with the original NT scheme, the method we develop evolves the numerical solution on a single grid; however ghost cells are implicitly used to avoid the resolution of the Riemann problems arising at the cell interfaces. We apply our method and solve classical one and two-dimensional unsteady shallow water problems. Our numerical results compare very well with those obtained using the original NT method, and are in good agreement with corresponding results appearing in the recent literature, thus confirming the efficiency and the potential of the proposed method.

Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems

Abstract 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.

Central finite volume schemes on nonuniform grids and applications

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.

Using gas-solid mixture conservation laws for volcanic eruptions

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.

Simulations on porous media with nanofluids–initial study

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.

Central schemes on unstructured grids

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.

Compressible gas-solid mixture conservation laws simulations

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.

Simulation of gas-liquid two-phase flow based on the Riemann problem

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.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...

Central Unstaggered Finite Volume Methods for Shallow Water Equations

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.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...