Kinetic Flux Vector Splitting Scheme Research Articles

Propagation of diffusing pollutant by kinetic flux-vector splitting method

Abstract In this article, the transport of a passive pollutant by a flow modeled by shallow water equations is numerically investigated. The kinetic flux-vector splitting (KFVS) scheme is extended to solve the one and two-dimensional equations. The first two equations of the considered model are mass and momentum equations and the third equation is the transport equation. The suggested scheme focuses on the direct splitting of the macroscopic flux functions at the cell interfaces. It achieves second-order accuracy by using MUSCL-type initial reconstruction and the Runge–Kutta time stepping technique. Several numerical test problems from literature are considered to check the efficiency and performance of the scheme. The results of the proposed scheme are compared to the central scheme for validation. It is found that the results of both the schemes are in close agreement with each other. However, our suggested KFVS scheme resolves the sharp discontinuous profiles precisely.

Numerical approximation of blood flow in arteries using kinetic flux-vector splitting (KFVS) scheme

This paper focuses on numerical simulations of blood flow in arteries with a variable stiffness and cross section at rest using a kinetic flux-vector splitting (KFVS) scheme. The fluxes are nonzero and exactly balanced by the source terms, and hence, the equations maintain the steady-state solutions. Our interest resides in the numerical estimate of the aforementioned flow model, whose complexity presents numerical difficulties. The suggested KFVS scheme suppresses the effects of numerical oscillations and dissipation. This makes the scheme more accurate and robust than the traditional numerical strategies. The KFVS scheme is then validated on examples taken from the literature. The central scheme has been used for the comparison of results obtained from proposed scheme.

Kinetic flux vector splitting scheme for solving non-reactive multi-component flows

This paper is about multi-component flow. There is no doubt that multi-component flow has a wide range of applications, specially in aerospace it plays a vital role during reentry of space ship into earth's atmosphere thats why it cannot be neglected for a proper vehicle design. In this paper one- and two-dimensional homogenous multi-component flow models are numerically investigated by using a high resolution splitting scheme and this scheme is known as Kinetic Flux Vector Splitting scheme. This scheme preserves positivity conditions and resolves shocks, rarefaction and contact discontinuity. The scheme is based on splitting of flux functions. Moreover Runge-Kutta time stepping technique with MUSCL-type initial reconstruction is used to guarantee higher order accurate solution. This work is first done by Qamar and Warnecke (2004) for the homogeneous multi-component flow equations using central scheme, here we investigate the same work using kinetic flux vector splitting scheme (KFVS) and compared the results with central scheme to verify the efficiency of studied scheme.

Mesoscopic-based finite volume solutions for waterhammer flows

ABSTRACTIt is becoming evident that high resolution finite volume (FV) numerical schemes for multi-dimensional waterhammer problems are needed for the development of an accurate transient-based condition assessment of pipelines. As a prerequisite, FV methods for one-dimensional waterhammer flows need to be developed. Such models can then be applied to multi-dimensional problems via directional splitting. In this paper, two FV schemes (one uses Bhatnagar–Gross–Krook Boltzmann (BGK) and the other uses kinetic flux vector splitting (KFVS) in the flux approximation) for one-dimensional waterhammer problems are formulated and applied. It is found that the KFVS and BGK schemes correctly capture discontinuity fronts in a classical reservoir-pipe-valve system. An oscillation-free collision time formulation has been proposed, tested and found to be robust. The stability of the proposed schemes is guaranteed when . Comparison between the BGK, KFVS, fixed-grid MOC, and first and second order Godunov schemes reveals that the second-order Godunov performs best, followed closely by the BGK scheme. However, the BGK should not be quickly dismissed since it is known that its real power becomes evident when dealing with complex physics, multi-scale and multi-dimensional problems.

High order finite-volume WENO scheme for five-equation model of compressible two-fluid flow

High order finite-volume weighted essentially non-oscillatory (WENO) scheme is applied for solving interface-capturing five-equation model of compressible two-fluid flows in one and two-space dimensions. The model is non-conservative and the governing equations consist of three equations, namely a continuity equation, a momentum equation and an energy balance equation for the fluid mixture and the remaining two are mass and energy equations for one of the two fluids. In the last equation, the non-conservative differential source term appears which is responsible for the energy exchange between fluids. The energy exchange is only due to mechanical work. The presence of non-conservative differential source terms in the two-fluid flow model introduce difficulties in developing high order accurate numerical schemes. The proposed numerical scheme is capable to preserve non-oscillatory property near strong discontinuities and gives high order accuracy in smooth regions. Different one and two-dimensional test problems are considered to analyze efficiency and accuracy of the proposed numerical algorithms. For validation, the solutions of proposed numerical scheme are compared with the results of already available high order kinetic flux-vector splitting scheme and discontinuous Galerkin scheme.

Application of kinetic flux vector splitting scheme for solving viscous quantum hydrodynamical model of semiconductor devices

Abstract In this article, one-dimensional viscous quantum hydrodynamical model of semiconductor devices is numerically investigated. The model treats the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behavior of electron flow in semiconductor devices. The nonlinear viscous quantum hydrodynamic models contain Euler-type equations for density and current, viscous and quantum correction terms, and a Poisson equation for electrostatic potential. Due to high nonlinearity of model equations, numerical solution techniques are applied to obtain their solutions. The proposed numerical scheme is a splitting scheme based on the kinetic flux-vector splitting (KFVS) method for the hyperbolic step, and a semi-implicit Runge-Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as device length, viscosities, different doping and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equations.

A kinetic flux vector splitting scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients.

This paper is concerned with the derivation of a well-balanced kinetic scheme to approximate a shallow flow model incorporating non-flat bottom topography and horizontal temperature gradients. The considered model equations, also called as Ripa system, are the non-homogeneous shallow water equations considering temperature gradients and non-uniform bottom topography. Due to the presence of temperature gradient terms, the steady state at rest is of primary interest from the physical point of view. However, capturing of this steady state is a challenging task for the applied numerical methods. The proposed well-balanced kinetic flux vector splitting (KFVS) scheme is non-oscillatory and second order accurate. The second order accuracy of the scheme is obtained by considering a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. The scheme is applied to solve the model equations in one and two space dimensions. Several numerical case studies are carried out to validate the proposed numerical algorithm. The numerical results obtained are compared with those of staggered central NT scheme. The results obtained are also in good agreement with the recently published results in the literature, verifying the potential, efficiency, accuracy and robustness of the suggested numerical scheme.

Application of Kinetic Flux Vector Splitting Scheme for Solving Viscous Quantum Hydrodynamical Model of Semiconductor Devices

In this article, one-dimensional viscous quantum hydro dynamical model of semiconductor devices is numerically investigated. The model treats the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behaviour of electron flow in semiconductor devices. The nonlinear viscous quantum hydrodynamic models contain Euler-type equations for density and current, viscous and quantum correction terms, and a Poisson equation for electrostatic potential. Due to high nonlinearity of model equations, numerical solution techniques are applied to obtain their solutions.. The proposed numerical scheme is a splitting scheme based on the kinetic flux-vector splitting (KFVS) method for the hyperbolic step, and a semi-implicit Runge-Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as device length, viscosities, different doping and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equation

Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations

This paper presents second-order accurate genuine BGK schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present schemes are derived from the analytical solution of the Anderson–Witting model, which is given for the first time and includes the “genuine” particle collisions in the gas transport process. The proposed schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

Application of kinetic flux vector splitting scheme for solving multi-dimensional hydrodynamical models of semiconductor devices

In this article, one and two-dimensional hydrodynamical models of semiconductor devices are numerically investigated. The models treat the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behavior of electron flow in semiconductor devices. Mathematically, the governing equations form a convection–diffusion type system with a right hand side describing the relaxation effects and interaction with a self consistent electric field. The proposed numerical scheme is a splitting scheme based on the kinetic flux-vector splitting (KFVS) method for the hyperbolic step, and a semi-implicit Runge–Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as low field mobility, device length, lattice temperature and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equations. A two dimensional simulation is also performed by KFVS method for a MESFET device, producing results in good agreement with those obtained by NT-central scheme.

Application of Central Upwind Scheme for Solving Special Relativistic Hydrodynamic Equations.

The accurate modeling of various features in high energy astrophysical scenarios requires the solution of the Einstein equations together with those of special relativistic hydrodynamics (SRHD). Such models are more complicated than the non-relativistic ones due to the nonlinear relations between the conserved and state variables. A high-resolution shock-capturing central upwind scheme is implemented to solve the given set of equations. The proposed technique uses the precise information of local propagation speeds to avoid the excessive numerical diffusion. The second order accuracy of the scheme is obtained with the use of MUSCL-type initial reconstruction and Runge-Kutta time stepping method. After a discussion of the equations solved and of the techniques employed, a series of one and two-dimensional test problems are carried out. To validate the method and assess its accuracy, the staggered central and the kinetic flux-vector splitting schemes are also applied to the same model. The scheme is robust and efficient. Its results are comparable to those obtained from the sophisticated algorithms, even in the case of highly relativistic two-dimensional test problems.

Central upwind scheme for a compressible two-phase flow model.

In this article, a compressible two-phase reduced five-equation flow model is numerically investigated. The model is non-conservative and the governing equations consist of two equations describing the conservation of mass, one for overall momentum and one for total energy. The fifth equation is the energy equation for one of the two phases and it includes source term on the right-hand side which represents the energy exchange between two fluids in the form of mechanical and thermodynamical work. For the numerical approximation of the model a high resolution central upwind scheme is implemented. This is a non-oscillatory upwind biased finite volume scheme which does not require a Riemann solver at each time step. Few numerical case studies of two-phase flows are presented. For validation and comparison, the same model is also solved by using kinetic flux-vector splitting (KFVS) and staggered central schemes. It was found that central upwind scheme produces comparable results to the KFVS scheme.

The space–time CE/SE method for solving one-dimensional special relativistic magnetohydrodynamics equations

The development of powerful computational tools for simulating phenomena that inhibits the relativistic magnetohydrodynamic (MHD) structure has become one of the core research issues in astrophysics and is one of the compelling fields. The relativistic MHD equations are more complex than the non-relativistic ones due to non-linear relations between conserved and state (primitive) variables. The non-linearity of the systems and the flow near speed of light pose major challenges to the theoretical investigation of the models and to the numerical solution techniques. The current numerical study is related to the implementation of the space–time conservation element and solution element (CE/SE) method for solving one-dimensional special relativistic magnetohydrodynamic (SRMHD) equations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building blocks of the suggested method. The method differs from previous techniques because of global and local flux conservation in a space–time domain without resorting to interpolation or extrapolation. For validation, the numerical results of the method are compared with the second order central and kinetic flux-vector splitting schemes. The one-dimensional computations of this paper verify the method’s efficiency, robustness and accuracy which are the key parameters in the context of astrophysical scenario.

Progress in gas-kinetic upwind schemes for the solution of Euler/Navier–Stokes equations – I: Overview

Three gas-kinetic upwind schemes for the solution of the Euler/Navier–Stokes equations are reviewed. They are Kinetic Flux-Vector Splitting (KFVS), Kinetic Wave/Particle Splitting (KWPS), and Bhatnagar–Gross–Krook (BGK) methods. For the Euler equations, the most sophisticated BGK scheme can be interpreted as a relaxation scheme between the two KFVS schemes with different moments, KFVS and KFVS_ u 0. It improves the accuracy over the KFVS scheme and the robustness over the KFVS_ u 0 scheme. The direct generalization of this relaxation approach to the Navier–Stokes equations leads to a much simpler BGK scheme than the one in the literature. In this simplified BGK scheme, there exist two types of particle collision time. The one in the BGK model acts as a relaxation parameter. Its role is to add some numerical dissipation from the KFVS scheme to the KFVS_ u 0 scheme. On the other hand, the one in the Chapman–Enskog expansion of the gas distribution function is related to physical dissipation. Following the same approach, another type of BGK schemes is further developed, which is a relaxation scheme between KWPS and KWPS_ u 0. In spite of the fact that the KWPS scheme is more diffusive than the KFVS scheme, a BGK scheme based on KWPS and KWPS_ u 0 is found not only computationally more efficient but also less diffusive than a BGK scheme based on KFVS and KFVS_ u 0. However, this issue needs further and more rigorous investigation by performing the numerical analysis of a model 1-D convection–diffusion equation.

On kinetic flux vector splitting schemes for quantum Euler equations

The kinetic flux vector splitting (KFVS) scheme, when used for quantum Euler equations, as was done by Yang et al [22], requires the integration of the quantum Maxwellian (Bose-Einstein or Fermi-Dirac distribution), giving a numerical flux much more complicated than the classical counterpart. As a result, a nonlinear 2 by 2 system that connects the macroscopic quantities temperature and fugacity with density and internal energy needs to be inverted by iterative methods at every spatial point and every time step. In this paper, we propose to use a simple classical KFVS scheme for the quantum hydrodynamics based on the key observation that the quantum and classical Euler equations share the same form if the (quantum) internal energy rather than temperature is used in the flux. This motivates us to use a classical Maxwellian - that depends on the internal energy rather than temperature - instead of the quantum one in the construction of the scheme, yielding a KFVS which is purely classical. This greatly simplifies the numerical algorithm and reduces the computational cost. The proposed schemes are tested on a 1-D shock tube problem for the Bose and Fermi gases in both classical and nearly degenerate regimes.

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge–Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space–time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.

Modified kinetic flux vector splitting schemes for compressible flows

We investigate the traditional kinetic flux vector splitting (KFVS) and BGK schemes for the compressible Euler equations. First, based on a careful study of the behavior of the discrete physical variables across the contact discontinuity, we analyze quantitatively the mechanism of inducing spurious oscillations of the velocity and pressure in the vicinity of the contact discontinuity for the first-order KFVS and BGK schemes. Then, with the help of this analysis, we propose a first-order modified KFVS (MKFVS) scheme which is oscillation-free in the vicinity of the contact discontinuity, provided certain consistent conditions are satisfied. Moreover, by using piecewise linear reconstruction and van Leer’s limiter, the first-order MKFVS scheme is extended to a second-order one, consequently, a nonoscillatory second-order MKFVS scheme is constructed. Finally, by combing the MKFVS schemes with the γ-model, we successfully extend the MKFVS schemes to multi-flows, and propose therefore a first- and second-order MKFVS schemes for multi-fluid computations, which are nonoscillatory across fluid interfaces. A number of numerical examples presented in this paper validate the theoretic analysis and demonstrate the good performance of the MKFVS schemes in simulation of contact discontinuities for both single- and multi-fluids.

A second‐order BGK scheme for the equations of radiation hydrodynamics

AbstractWe present a second‐order BGK scheme for the equations of multidimensional radiation hydrodynamics (RHE) in zero diffusion limit by using the generalized Maxwellian (Comput. Fluids 2000; 29:917–933) and the second‐order BGK scheme for the Euler equations (J. Comput. Phys. 1993; 109:53–66; Gas Kinetic Scheme for Unsteady Compressible Flow Simulations. Lecture Note Series 1998‐03. Von Kárman Institute for Fluid Dynamics, 1998). This extends a (first‐order) kinetic flux vector splitting scheme (KFVS) for RHE (Tang and Wu (2000)) to a second‐order BGK scheme. Several one‐ and two‐dimensional numerical examples demonstrate improvement of the scheme in accuracy and resolution compared with the KFVS scheme (Tang and Wu (2000)) and the first‐order BGK scheme. Copyright © 2006 John Wiley & Sons, Ltd.

The Runge-Kutta DG finite element method and the KFVS scheme for compressible flow simulations

This article presents a new type of second-order scheme for solving the system of Euler equations, which combines the Runge-Kutta discontinuous Galerkin (DG) finite element method and the kinetic flux vector splitting (KFVS) scheme. We first discretize the Euler equations in space with the DG method and then the resulting system from the method-of-lines approach will be discretized using a Runge-Kutta method. Finally, a second-order KFVS method is used to construct the numerical flux. The proposed scheme preserves the main advantages of the DG finite element method including its flexibility in handling irregular solution domains and in parallelization. The efficiency and effectiveness of the proposed method are illustrated by several numerical examples in one- and two-dimensional spaces. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006