On the asymptotic properties of IMEX Runge-Kutta schemes for hyperbolic balance laws

Implicit-Explicit (IMEX) schemes are a powerful tool in the development of numerical methods for hyperbolic systems with stiff sources. Here we focus our attention on the asymptotic properties of such schemes, like the preservation of steady-states (well-balanced property) and the behavior in presence of small space-time scales (asymptotic preservation property). We analyze conditions under which the standard additive approach based on taking the fluxes explicitly and the sources implicitly yields a well-balanced behavior. In addition, we consider a partitioned strategy which possesses better well-balanced properties. The behavior of the additive and partitioned approaches under classical scaling limits is then studied in the context of asymptotic-preserving schemes. Additional order conditions that guarantee the correct behavior of the schemes in the Navier-Stokes regime are derived. Several examples illustrate these asymptotic behaviors and the performance of the new methods.

[1]  SEBASTIANO BOSCARINO.  Error Analysis of IMEX Runge-Kutta Methods Derived from Differential-Algebraic Systems , 2007, SIAM J. Numer. Anal..

[2]  Giacomo Dimarco,et al.  Asymptotic Preserving Implicit-Explicit Runge-Kutta Methods for Nonlinear Kinetic Equations , 2012, SIAM J. Numer. Anal..

[3]  Inmaculada Higueras,et al.  Strong Stability for Additive Runge-Kutta Methods , 2006, SIAM J. Numer. Anal..

[4]  A. Klar.  An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit , 1998 .

[5]  Laurent Gosse,et al.  Space Localization and Well-Balanced Schemes for Discrete Kinetic Models in Diffusive Regimes , 2003, SIAM J. Numer. Anal..

[6]  Giovanni Russo,et al.  Flux-Explicit IMEX Runge-Kutta Schemes for Hyperbolic to Parabolic Relaxation Problems , 2013, SIAM J. Numer. Anal..

[7]  C. D. Levermore,et al.  Hyperbolic conservation laws with stiff relaxation terms and entropy , 1994 .

[8]  Adrian Sandu,et al.  A Generalized-Structure Approach to Additive Runge-Kutta Methods , 2015, SIAM J. Numer. Anal..

[9]  G. Russo,et al.  Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations , 2000 .

[10]  Jagdish Chandra.  Quasilinear Hyperbolic Systems and Waves (A. Jeffrey) , 1978 .

[11]  Lorenzo Pareschi,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2010, 1009.2757.

[12]  Shi Jin.  Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1995 .

[13]  Laurent Gosse.  Sur la stabilité des approximations implicites des lois de conservation scalaires non homogènes , 1999 .

[14]  J. Brandts.  [Review of: W. Hundsdorfer, J.G. Verwer (2003) Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations] , 2006 .

[15]  Laurent Gosse,et al.  Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes , 1996 .

[16]  Xiaolin Zhong,et al.  Additive Semi-Implicit Runge-Kutta Methods for Computing High-Speed Nonequilibrium Reactive Flows , 1996 .

[17]  Lorenzo Pareschi,et al.  Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit , 2013, SIAM J. Sci. Comput..

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

[19]  Giacomo Dimarco,et al.  Numerical methods for kinetic equations* , 2014, Acta Numerica.

[20]  M. Carpenter,et al.  Additive Runge-Kutta Schemes for Convection-Diffusion-Reaction Equations , 2003 .

[21]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[22]  G. Russo,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2005 .

[23]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[24]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

[25]  Tong Wu,et al.  Steady State and Sign Preserving Semi-Implicit Runge-Kutta Methods for ODEs with Stiff Damping Term , 2015, SIAM J. Numer. Anal..

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

[27]  Adrian Sandu,et al.  A Class Of Implicit-Explicit Two-Step Runge-Kutta Methods , 2015, SIAM J. Numer. Anal..

[28]  Giovanni Russo,et al.  Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation , 1997 .

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

[30]  Sebastiano Boscarino,et al.  On an accurate third order implicit-explicit Runge--Kutta method for stiff problems , 2009 .

[31]  Randall J. LeVeque,et al.  Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods , 1998 .

[32]  Willem Hundsdorfer,et al.  IMEX extensions of linear multistep methods with general monotonicity and boundedness properties , 2007, J. Comput. Phys..

[33]  G. Russo,et al.  High Order Asymptotically Strong-Stability-Preserving Methods for Hyperbolic Systems with Stiff Relaxation , 2003 .

[34]  Shi Jin,et al.  Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations , 1999, SIAM J. Sci. Comput..

[35]  Laurent Gosse,et al.  A Well-Balanced Scheme Using Non-Conservative Products Designed for Hyperbolic Systems of Conservati , 2001 .

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

[37]  Jan G. Verwer,et al.  An Implicit-Explicit Runge-Kutta-Chebyshev Scheme for Diffusion-Reaction Equations , 2004, SIAM J. Sci. Comput..

[38]  Chi-Wang Shu,et al.  High-order well-balanced schemes and applications to non-equilibrium flow , 2009, J. Comput. Phys..

[39]  Yulong Xing,et al.  High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms , 2006, J. Sci. Comput..

[40]  Lorenzo Pareschi,et al.  Central Differencing Based Numerical Schemes for Hyperbolic Conservation Laws with Relaxation Terms , 2001, SIAM J. Numer. Anal..

[41]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[42]  Giovanni Russo,et al.  On a Class of Uniformly Accurate IMEX Runge--Kutta Schemes and Applications to Hyperbolic Systems with Relaxation , 2009, SIAM J. Sci. Comput..

[43]  Randall J. LeVeque,et al.  A study of numerical methods for hyperbolic conservation laws with stiff source terms , 1990 .

[44]  Lorenzo Pareschi,et al.  Numerical schemes for kinetic equations in diffusive regimes , 1998 .

[45]  Inmaculada Higueras,et al.  Characterizing Strong Stability Preserving Additive Runge-Kutta Methods , 2009, J. Sci. Comput..