Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes

We prove the uniform AP convergence on unstructured meshes in 2D of a generalization, of the Gosse-Toscani 1D scheme for the hyperbolic heat equation. This scheme is also a nodal extension in 2D of the Jin-Levermore scheme described in [18] for the 1D case. In 2D, the proof is performed using a new diffusion scheme.

[1]  Pierre Charrier,et al.  An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions , 2007, J. Sci. Comput..

[2]  François Golse,et al.  The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method , 1999 .

[3]  OIS GOLSEy,et al.  THE CONVERGENCE OF NUMERICAL TRANSFER SCHEMES IN DIFFUSIVE REGIMES I: DISCRETE-ORDINATE METHOD FRANC , 2007 .

[4]  I. Bihari.  A generalization of a lemma of bellman and its application to uniqueness problems of differential equations , 1956 .

[5]  Francis Filbet,et al.  Analysis of an Asymptotic Preserving Scheme for Relaxation Systems , 2011 .

[6]  E. Franck,et al.  An asymptotic preserving scheme with the maximum principle for the model on distorded meshes , 2012 .

[7]  Lorenzo Pareschi,et al.  Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations , 1998 .

[8]  Philippe G. LeFloch,et al.  Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations , 2010, Math. Comput..

[9]  Shi Jin,et al.  The discrete-ordinate method in diffusive regimes , 1991 .

[10]  E. Franck,et al.  Asymptotic Preserving Finite Volumes Discretization For Non-Linear Moment Model On Unstructured Meshes , 2011 .

[11]  Laurent Gosse,et al.  Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Case's elementary solutions , 2011 .

[12]  Shi Jin.  ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW , 2010 .

[13]  Bruno Després.  Weak consistency of the cell-centered Lagrangian GLACE scheme on general meshes in any dimension , 2010 .

[14]  Laurent Gosse,et al.  A Two-Dimensional Version of the Godunov Scheme for Scalar Balance Laws , 2014, SIAM J. Numer. Anal..

[15]  Nicolas Crouseilles,et al.  A dynamic multi-scale model for transient radiative transfer calculations , 2013 .

[16]  Luc Mieussens,et al.  A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit , 2008, SIAM J. Sci. Comput..

[17]  Lorenzo Pareschi,et al.  Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation , 2000, SIAM J. Numer. Anal..

[18]  Laurent Gosse,et al.  An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations , 2002 .

[19]  Luc Mieussens,et al.  Analysis of an Asymptotic Preserving Scheme for Linear Kinetic Equations in the Diffusion Limit , 2009, SIAM J. Numer. Anal..

[20]  G. Toscani,et al.  Diiusive Relaxation Schemes for Discrete-velocity Kinetic Equations , 2007 .

[21]  Thierry Gallouët,et al.  Finite volume method , 2010, Scholarpedia.

[22]  Shi Jin,et al.  Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations , 2000, SIAM J. Numer. Anal..

[23]  N. Crouseilles,et al.  An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits. , 2011 .

[24]  Christophe Berthon,et al.  Asymptotic preserving HLL schemes , 2011 .

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

[26]  S. Mancini,et al.  DIFFUSION LIMIT OF THE LORENTZ MODEL: ASYMPTOTIC PRESERVING SCHEMES , 2002 .

[27]  C. D. Levermore,et al.  Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1996 .

[28]  Bruno Després,et al.  Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes , 2012, Numerische Mathematik.

[29]  Bruno Després,et al.  Lax theorem and finite volume schemes , 2003, Math. Comput..

[30]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.