TESTING IMPROVEMENTS OF A WELL-BALANCED METHOD FOR THE MODEL OF A FLUID IN A NOZZLE WITH VARIABLE CROSS-SECTION

A set of improvements of a well-balanced scheme for the model of fluid flows in a nozzle with variable cross-section is presented. Relying on the well-balanced method introduced in our earlier work, we use the steady state solutions to absorb the nonconservative term. The underlying numerical fluxes operating on these steady states are obtained as convex combinations of the numerical fluxes of a first-order and a second-order schemes. The improvements are still well-balanced schemes. Then, we present many numerical tests, which establishes the efficiency of these numerical schemes. These schemes can provide us with very desirable approximations for any initial data: data in supersonic or subsonic regions, and data in both of these two kinds of regions. All the tests also show that the accuracy of the method by the improvements is improved.

[1]  D. Marchesin,et al.  A RIEMANN PROBLEM IN GAS DYNAMICS WITH BIFURCATION , 1986 .

[2]  P. Floch.  Shock Waves for Nonlinear Hyperbolic Systems in Nonconservative Form , 1989 .

[3]  G. D. Maso,et al.  Definition and weak stability of nonconservative products , 1995 .

[4]  Blake Temple,et al.  Convergence of the 2×2 Godunov Method for a General Resonant Nonlinear Balance Law , 1995, SIAM J. Appl. Math..

[5]  J. Greenberg,et al.  Analysis and Approximation of Conservation Laws with Source Terms , 1997 .

[6]  R. Abgrall,et al.  A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows , 1999 .

[7]  L. Gosse.  A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms☆ , 2000 .

[8]  Ramaz Botchorishvili,et al.  Equilibrium schemes for scalar conservation laws with stiff sources , 2003, Math. Comput..

[9]  M. Thanh,et al.  The Riemann Problem for Fluid Flows in a Nozzle with Discontinuous Cross-Section , 2003 .

[10]  O. Pironneau,et al.  Finite volume schemes with equilibrium type discretization of source terms for scalar conservation laws , 2003 .

[11]  T. Gallouët,et al.  Numerical modeling of two-phase flows using the two-fluid two-pressure approach , 2004 .

[12]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[13]  Gerald Warnecke,et al.  On the Solution to The Riemann Problem for the Compressible Duct Flow , 2004, SIAM J. Appl. Math..

[14]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[15]  A. Leroux,et al.  A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon , 2004 .

[16]  Mai Duc Thanh,et al.  Numerical Solutions to Compressible Flows in a Nozzle with Variable Cross-section , 2005, SIAM J. Numer. Anal..

[17]  M. Lallemand,et al.  Pressure relaxation procedures for multiphase compressible flows , 2005 .

[18]  Shi Jin,et al.  Two Interface-Type Numerical Methods for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations , 2005, SIAM J. Sci. Comput..

[19]  Manuel Jesús Castro Díaz,et al.  High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems , 2006, Math. Comput..

[20]  C. Parés.  Numerical methods for nonconservative hyperbolic systems: a theoretical framework. , 2006 .

[21]  M. Thanh,et al.  The Riemann problem for the shallow water equations with discontinuous topography , 2007, 0712.3778.

[22]  Manuel Jesús Castro Díaz,et al.  Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes , 2008, J. Comput. Phys..

[23]  D. Kröner,et al.  The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section , 2008 .

[24]  C. Chalons,et al.  Relaxation and numerical approximation of a two-fluid two-pressure diphasic model , 2009 .

[25]  Mai Duc Thanh,et al.  The Riemann Problem for a Nonisentropic Fluid in a Nozzle with Discontinuous Cross-Sectional Area , 2009, SIAM J. Appl. Math..

[26]  M. J. Castro,et al.  ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows , 2009 .

[27]  M. J. Castro,et al.  FORCE schemes on unstructured meshes II: Non-conservative hyperbolic systems , 2010 .

[28]  Nguyen Thanh Nam,et al.  Numerical approximation for a Baer-Nunziato model of two-phase flows , 2011 .

[29]  W. Bussière,et al.  Unsteady compressible flow in ducts with varying cross-section : comparison between the nonconservative Euler system and the axisymmetric flow model , 2012 .

[30]  Christophe Chalons,et al.  A robust numerical method for approximating solutions of a model of two-phase flows and its properties , 2012, Appl. Math. Comput..

[31]  D. Kröner,et al.  Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable cross-section , 2012 .

[32]  Mai Duc Thanh,et al.  Numerical treatment in resonant regime for shallow water equations with discontinuous topography , 2013, Commun. Nonlinear Sci. Numer. Simul..

[33]  Mai Duc Thanh,et al.  Building fast well-balanced two-stage numerical schemes for a model of two-phase flows , 2014, Commun. Nonlinear Sci. Numer. Simul..

[34]  M. Thanh.  WELL-BALANCED ROE-TYPE NUMERICAL SCHEME FOR A MODEL OF TWO-PHASE COMPRESSIBLE FLOWS , 2014 .