An IMEX Method for the Euler Equations that Posses Strong Non-Linear Heat Conduction and Stiff Source Terms (Radiation Hydrodynamics)

Here, we present a truly second order time accurate self-consistent IMEX (IMplicit/EXplicit) method for solving the Euler equations that posses strong nonlinear heat conduction and very stiff source terms (Radiation hydrodynamics). This study essentially summarizes our previous and current research related to this subject (Kadioglu & Knoll, 2010; 2011; Kadioglu, Knoll & Lowrie, 2010; Kadioglu, Knoll, Lowrie & Rauenzahn, 2010; Kadioglu et al., 2009; Kadioglu, Knoll, Sussman M 1995; Bates et al., 2001; Kadioglu & Knoll, 2010; 2011; Kadioglu, Knoll, Lowrie & Rauenzahn, 2010; Kadioglu et al., 2009; Khan & Liu, 1994; Kim & Moin, 1985; Lowrie et al., 1999; Ruuth, 1995). These methods are particularly attractive when dealing with physical systems that consist of multiple physics (multi-physics problems such as coupling of neutron dynamics to thermal-hydrolic or to thermal-mechanics in reactors) or fluid dynamics problems that exhibit multiple time scales such as advection-diffusion, reaction-diffusion, or advection-diffusion-reaction problems. In general, governing equations for these kinds of systems consist of stiff and non-stiff terms. This poses numerical challenges in regards to time integrations, since most of the temporal numerical methods are designed specific for either stiff or non-stiff problems. Numerical methods that can handle both physical behaviors are often referred to as IMEX methods. A typical IMEX method isolates the stiff and non-stiff parts of the governing system and employs an explicit discretization strategy that solves the non-stiff part and an implicit technique that solves the stiff part of the problem. This standard IMEX approach can be summarized by considering a simple prototype model. Let us consider the following scalar model ut = f (u) + g(u), (1)

[1]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[2]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[3]  Jim E. Morel,et al.  The Coupling of Radiation and Hydrodynamics , 1999 .

[4]  Paul R. Woodward,et al.  Numerical Simulations for Radiation Hydrodynamics. I. Diffusion Limit , 1998 .

[5]  R. E. Marshak,et al.  Effect of Radiation on Shock Wave Behavior , 1958 .

[6]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[7]  Liaqat Ali Khan,et al.  An operator splitting algorithm for coupled one-dimensional advection-diffusion-reaction equations , 1995 .

[8]  Rick M. Rauenzahn,et al.  Radiative shock solutions in the equilibrium diffusion limit , 2007 .

[9]  Paul R. Woodward,et al.  Numerical Simulations for Radiation Hydrodynamics , 2000 .

[10]  Steven J. Ruuth.  Implicit-explicit methods for reaction-diffusion problems in pattern formation , 1995 .

[11]  D. Mihalas,et al.  Foundations of Radiation Hydrodynamics , 1985 .

[12]  Stanley Osher,et al.  A second order primitive preconditioner for solving all speed multi-phase flows , 2005 .

[13]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

[14]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[15]  William J. Rider,et al.  Time Step Size Selection for Radiation Diffusion Calculations , 1999 .

[16]  William J. Rider,et al.  On consistent time-integration methods for radiation hydrodynamics in the equilibrium diffusion limit: low-energy-density regime , 2001 .

[17]  Rick M. Rauenzahn,et al.  A second order self-consistent IMEX method for radiation hydrodynamics , 2010, J. Comput. Phys..

[18]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[19]  M. Sussman,et al.  A Second Order JFNK-based IMEX Method for Single and Multi-phase Flows , 2011 .

[20]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[21]  Yousef Saad,et al.  Hybrid Krylov Methods for Nonlinear Systems of Equations , 1990, SIAM J. Sci. Comput..

[22]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[23]  Robert B. Lowrie,et al.  Radiative shock solutions with grey nonequilibrium diffusion , 2008 .

[24]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[25]  L. Ensman,et al.  Test problems for radiation and radiation-hydrodynamics codes , 1994 .

[26]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[27]  R. P. Drake.  Theory of radiative shocks in optically thick media , 2007 .

[28]  Dana A. Knoll,et al.  Multiphysics Analysis of Spherical Fast Burst Reactors , 2009 .

[29]  G. C. Pomraning.  The Equations of Radiation Hydrodynamics , 2005 .

[30]  R. Bowers,et al.  Numerical Modeling in Applied Physics and Astrophysics , 1991 .

[31]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[32]  Steven J. Ruuth,et al.  Implicit-Explicit Methods for Time-Dependent PDE''s , 1993 .

[33]  Dana A. Knoll,et al.  A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems , 2010, J. Comput. Phys..

[34]  J. Anderson,et al.  Modern Compressible Flow , 2012 .

[35]  J. Castor.  Radiation Hydrodynamics: Examples , 2004 .

[36]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[37]  R. Wolke,et al.  Strang Splitting Versus Implicit-Explicit Methods in Solving Chemistry Transport Models A contribution to subproject GLOREAM , 1999 .

[38]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .