Constructing a Godunov-type scheme for the model of a general fluid flow in a nozzle with variable cross-section

A Godunov-type scheme for the model of a general fluid flow in a nozzle with variable cross-section is presented. The model has the form of a nonconservative system of balance laws, which poses many challenging questions for study. First, exact Riemann solvers in computing form are described in both subsonic and supersonic regions. Second, the computable exact solutions of local Riemann problem are incorporated into a Godunov-type scheme. Third, the scheme is shown to be well-balanced in the sense that it can capture exactly stationary waves. Finally, numerical tests for data belong to both subsonic and supersonic regions are presented. These tests show that the scheme has a very fine accuracy. Especially, the scheme can give very good approximations to the exact solutions even in the resonant phenomenon where the solution contains three waves of the same speed.

[1]  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..

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

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

[4]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[5]  J. Greenberg,et al.  A well-balanced scheme for the numerical processing of source terms in hyperbolic equations , 1996 .

[6]  Eleuterio F. Toro,et al.  Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry , 2008, J. Comput. Phys..

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

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

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

[10]  P. Goatin,et al.  The Riemann problem for a class of resonant hyperbolic systems of balance laws , 2004 .

[11]  E. Isaacson,et al.  Nonlinear resonance in systems of conservation laws , 1992 .

[12]  Philippe G. LeFloch,et al.  A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime , 2011, J. Comput. Phys..

[13]  Eleuterio F. Toro,et al.  On some fast well-balanced first order solvers for nonconservative systems , 2009, Math. Comput..

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

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

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

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

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

[19]  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..

[20]  Giorgio Rosatti,et al.  The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations , 2010, J. Comput. Phys..

[21]  Smadar Karni,et al.  A Hybrid Algorithm for the Baer-Nunziato Model Using the Riemann Invariants , 2010, J. Sci. Comput..

[22]  On a General Definition of the Godunov Method for Nonconservative Hyperbolic Systems. Application to Linear Balance Laws , 2006 .

[23]  Alfredo Bermúdez,et al.  Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines , 2016, J. Comput. Phys..

[24]  D. Kröner,et al.  TESTING IMPROVEMENTS OF A WELL-BALANCED METHOD FOR THE MODEL OF A FLUID IN A NOZZLE WITH VARIABLE CROSS-SECTION , 2014 .

[25]  P. Raviart,et al.  A Godunov-type method for the seven-equation model of compressible two-phase flow , 2012 .

[26]  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..

[27]  S. T. Munkejord.  Comparison of Roe-type methods for solving the two-fluid model with and without pressure relaxation , 2007 .

[28]  Eleuterio F. Toro,et al.  Towards Very High Order Godunov Schemes , 2001 .

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

[30]  Shi Jin,et al.  AN EFFICIENT METHOD FOR COMPUTING HYPERBOLIC SYSTEMS WITH GEOMETRICAL SOURCE TERMS HAVING CONCENTRATIONS ∗1) , 2004 .

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

[32]  Donald W. Schwendeman,et al.  The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow , 2006, J. Comput. Phys..

[33]  Mai Duc Thanh,et al.  A Godunov-type scheme for the isentropic model of a fluid flow in a nozzle with variable cross-section , 2015, Appl. Math. Comput..

[34]  Michael Dumbser,et al.  Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms , 2012, J. Comput. Phys..

[35]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

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

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

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

[39]  M. Thanh.  A phase decomposition approach and the Riemann problem for a model of two-phase flows , 2014 .

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

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