Kinetic Flux Vector Splitting Research Articles

A central-upwind scheme for two-phase shallow granular flow model

In this article, a central-upwind scheme (CUP) is extended to solve two-phase shallow granular flow (TPSF) model. The flow in this case is considered as incompressible and it is regarded to be a shallow layer of granular and liquid material. The model consists of continuity and momentum equations for both solid and liquid phases. Our main intrigue is the numerical approximation of the above-mentioned solid-liquid model, the complexity of which raises numerical challenges. The proposed method is non-oscillating upwind biased that does not require a Riemann solver at each step. A few test problems are approximated numerically to check the performance of suggested scheme. For validation the outcomes are compared with the conservation element solution element (CE/SE) and kinetic flux vector splitting (KFVS) schemes.

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.

A hybrid kinetic WGVC–WENO scheme for compressible flows

In this paper, we propose a hybrid kinetic weighted group velocity control–weighted essentially non-oscillatory (WGVC–WENO) scheme which combines the WGVC–WENO reconstruction technique He et al. (2014) with the hybrid kinetic method proposed in Liu et al. (2015). The new scheme is based on the idea of hybridization. The hybrid kinetic method is adopted for flux calculations, and the WGVC–WENO technique is adopted for physical reconstructions. In smooth flow regions, the collision-related flux method and WGVC reconstruction technique play a leading role. The kinetic flux vector splitting (KFVS) method and WENO reconstruction technique are dominant in discontinuous regions. The third-order TVD Runge–Kutta method is used for temporal discretization. Many one-dimensional and two-dimensional numerical examples show that the new scheme has the properties of high accuracy, low dissipation, and good shock-capturing ability.

An implicit kinetic inviscid flux for predicting continuum flows in all speed regimes

In this study, the kinetic inviscid flux (KIF) is improved and coupled with an implicit strategy. The KIF is a recently proposed numerical method, which is a dynamic combination of the kinetic flux vector splitting (KFVS) method and the totally thermalized transport (TTT) method. The inherent microscopic mechanism of the KFVS makes the KIF good at solving shock waves and avoiding the numerical shock instability phenomenon. When developing the implicit KIF, it is noticed that, in boundary layers, the KFVS part of the KIF not only reduces the accuracy but also seriously reduces the Courant–Friedrichs–Lewy (CFL) number. As a result, a new weight is proposed in this paper to combine the KFVS method with the TTT method properly. Besides admitting the use of larger CFL numbers, this new weight also contributes to more accurate numerical results like pressure, friction coefficient, and heat flux when solving shock waves, boundary layers, and complex supersonic/hypersonic flows. To examine the validity, accuracy, and efficiency of the proposed method, six numerical test cases covering the whole speed regime are conducted, including the hypersonic viscous flow past a cylinder, the hypersonic double-cone flow, the hypersonic double-ellipsoid flow, the laminar shock-boundary layer interaction, the supersonic flow around a ramp segment and the subsonic lid-driven cavity flow.

High-order kinetic flow solver based on the flux reconstruction framework.

The goal of this paper is to develop a high-order numerical method based on the kinetic inviscid flux (KIF) method and flux reconstruction (FR) framework. The KIF aims to find a balance between the excellent merits of the gas-kinetic scheme (GKS) and the lower computational costs. The idea of KIF can be viewed as an inviscid-viscous splitting version of the gas-kinetic scheme, and Shu and Ohwada have made the fundamental contribution. The combination of totally thermalized transport (TTT) scheme and kinetic flux vector splitting (KFVS) method are achieved in KIF. Using a coefficient which is related to time step and averaged collision time, KIF can adjust the weights of TTT and KFVS flux in the simulation adaptively. By doing the inviscid-viscous splitting, KIF is very suitable and easy to integrate into the existing framework. The well-understood FR framework is used widely for the advantages of robustness, economical costs, and compactness. The combination of KIF and FR is originated by three motivations. The first purpose is to develop a high-order method based on the gas-kinetic theory. The second reason is to keep the advantages of GKS. The last aim is that the designed method should be more efficient. In present work, we use the KIF method to replace the Riemann flux solver applied in the interfaces of elements. The common solution at the interface is computed according to the gas-kinetic theory, which makes the combination of KIF and FR scheme more reasonable and available. The accuracy and performance of present method are validated by several numerical cases. The Taylor-Green vortex problem has been used to verify its potential to simulate turbulent flows.

Multiscale kinetic inviscid flux extracted from a gas-kinetic scheme for simulating incompressible and compressible flows.

A kinetic inviscid flux (KIF) is proposed for simulating incompressible and compressible flows. It is constructed based on the direct modeling of multiscale flow behaviors, which is used in the gas-kinetic scheme (GKS), the unified gas-kinetic scheme (UGKS), the discrete unified gas-kinetic scheme (DUGKS), etc. In KIF, the discontinuities (such as the shock wave) that cannot be well resolved by mesh cells are mainly solved by the kinetic flux vector splitting (KFVS) method representing the free transport mechanism (or microscale mechanism), while in other flow regions that are smooth, the flow behavior is solved mainly by the central-scheme-like totally thermalized transport (TTT). The weights of KFVS and TTT in KIF is automatically determined by those in the theory of direct modeling. Two ways of choosing the weights in KIF are proposed, which are actually the weights adopted in the UGKS and the DUGKS, respectively. By using the test cases of the Sod shock tube, the rarefaction wave, the boundary layer of a flat plate, the cavity flow, hypersonic flow over a circular cylinder, the shock and turbulent boundary iteration, and transonic flow over a three-dimensional M6 wing, the validity and accuracy of the present method are examined. The KIF does not suffer from the carbuncle phenomenon, and does not introduce extra numerical viscosity in smooth regions. Especially in the case of hypersonic cylinder, it gives quite sharp and clear density and temperature contours. The KIF can be viewed as an inviscid-viscous splitting version of the GKS. By doing this splitting, it is easy to be used in traditional computational fluid dynamics frameworks. It can also be classified as a type in the numerical schemes based on the kinetic theory that are represented by the works of Sun etal. [Adv. Appl. Math. Mech. 8, 703 (2016)10.4208/aamm.2015.m1071] and Ohwada etal. [J. Comput. Phys. 362, 131 (2018)JCTPAH0021-999110.1016/j.jcp.2018.02.019], except the weights are determined by the weights of direct modeling.

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.

Third-order WENO schemes with kinetic flux vector splitting

The numerical methods for solving the hyperbolic conservation laws are mainly based on the upwinding principle. The weighted essentially nonoscillatory (WENO) schemes are among the most commonly used numerical schemes for solving the hyperbolic conservation laws due to their nonoscillatory property and high order good shock capturing abilities. The upwinding principle requires the splitting of the flux into its positive and negative parts. WENO schemes have so far been widely implemented using the Lax Friedrich flux vector splitting technique for splitting purpose. In this article, we have implemented them with the kinetic flux vector splitting (KFVS) technique. The comparison has been made with both flux vector splitting techniques for one dimensional Euler equations. The results obtained are as expected, considerable improvement has been seen in the solution obtained by the scheme which has been implemented with the kinetic flux vector splitting. The reason being KFVS is based on the sign of the intermolecular particle velocity, which is a physical quantity.

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.

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.

Numerical solution of special ultra-relativistic Euler equations using central upwind scheme

Abstract This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.

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.

Time-implicit gas-kinetic scheme

Abstract The construction of time-implicit gas-kinetic scheme (GKS) is investigated to enhance its efficiency in engineering applications, especially with large time scales. GKS is based on BGK equation to simulate compressible flows, which describes particles evolution with mesoscopic distribution function. To develop an implicit GKS, classical time-implicit techniques can be directly adopted, with the employment of upwind approximate factorization for Euler equations. However, the characteristic relation of BGK equation is quite different to that of macroscopic compressible Euler/Navier-Stokes equations, which may decrease the gain from implicit techniques, especially in high-speed flows. To further improve the performance, the preconditioned generalized minimum residual (GMRES) based on numerical Jacobian matrix is considered, and the kinetic flux vector splitting (KFVS) is suggested to generate the precondition matrix. The proposed new time-implicit GKS, GMRES-GKS shows great improvements in temporal accuracy, resolution and convergence in several typical high-speed benchmark tests, ranging from transonic to hypersonic, steady to unsteady and inviscid to turbulent flows.

A Switch Function-Based Gas-Kinetic Scheme for Simulation of Inviscid and Viscous Compressible Flows

AbstractIn this paper, a switch function-based gas-kinetic scheme (SF-GKS) is presented for the simulation of inviscid and viscous compressible flows. With the finite volume discretization, Euler and Navier-Stokes equations are solved and the SF-GKS is applied to evaluate the inviscid flux at cell interface. The viscous flux is obtained by the conventional smooth function approximation. Unlike the traditional gas-kinetic scheme in the calculation of inviscid flux such as Kinetic Flux Vector Splitting (KFVS), the numerical dissipation is controlled with a switch function in the present scheme. That is, the numerical dissipation is only introduced in the region around strong shock waves. As a consequence, the present SF-GKS can well capture strong shock waves and thin boundary layers simultaneously. The present SF-GKS is firstly validated by its application to the inviscid flow problems, including 1-D Euler shock tube, regular shock reflection and double Mach reflection. Then, SF-GKS is extended to solve viscous transonic and hypersonic flow problems. Good agreement between the present results and those in the literature verifies the accuracy and robustness of SF-GKS.

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.