In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamicall...In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.展开更多
In this paper we proposed a modified Baer-Nunziato model for compressible multi-fluid flows,with main attention on the energy exchange between the two fluids.The proposed model consists of eleven PDEs;however,the use ...In this paper we proposed a modified Baer-Nunziato model for compressible multi-fluid flows,with main attention on the energy exchange between the two fluids.The proposed model consists of eleven PDEs;however,the use of the particular phase evolving variables may reduce the model to have only six PDEs.The main advantage of the model is that the Abgrall’s UPV criterion on mixture velocity and pressure is satisfied without affecting either its hyperbolicity or its conservations of the two individual masses,momentum or total energy.An Lax-Friedrichs scheme is built for a particular case of the proposed model.When the two fluids in the fluid mixture are both of the linear Mie-Gruneisen type,the scheme satisfies the Abgrall’s UPV criterion on mixture velocity and pressure.Numerical experiments with polytropic,barotropic,stiffened and van der Waals fluids show that the scheme is efficient and able to treat fluids characterized with quite different thermodynamics.展开更多
文摘In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.
基金supported by China National Science Foundation Grant No.10971132Leading Academic Discipline Project of Shanghai Municipal Education Commission No.J50101 and Shanghai Pu Jiang program[2006]118.
文摘In this paper we proposed a modified Baer-Nunziato model for compressible multi-fluid flows,with main attention on the energy exchange between the two fluids.The proposed model consists of eleven PDEs;however,the use of the particular phase evolving variables may reduce the model to have only six PDEs.The main advantage of the model is that the Abgrall’s UPV criterion on mixture velocity and pressure is satisfied without affecting either its hyperbolicity or its conservations of the two individual masses,momentum or total energy.An Lax-Friedrichs scheme is built for a particular case of the proposed model.When the two fluids in the fluid mixture are both of the linear Mie-Gruneisen type,the scheme satisfies the Abgrall’s UPV criterion on mixture velocity and pressure.Numerical experiments with polytropic,barotropic,stiffened and van der Waals fluids show that the scheme is efficient and able to treat fluids characterized with quite different thermodynamics.
