Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method

In this paper we derive a priori and a posteriori error estimates for cell centered finite volume approximations of nonlinear conservation laws on polygonal bounded domains. Numerical experiments show the applicability of the a posteriori result for the derivation of local adaptive solution strategies.

[1]  Endre Süli,et al.  Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity , 1996 .

[2]  Laurent Gosse,et al.  Two A Posteriori Error Estimates for One-Dimensional Scalar Conservation Laws , 2000, SIAM J. Numer. Anal..

[3]  E. Tadmor.  Local error estimates for discontinuous solutions of nonlinear hyperbolic equations , 1991 .

[4]  S. Kružkov.  FIRST ORDER QUASILINEAR EQUATIONS IN SEVERAL INDEPENDENT VARIABLES , 1970 .

[5]  Alexander Kurganov,et al.  A Smoothness Indicator for Adaptive Algorithms for Hyperbolic Systems , 2002 .

[6]  D. Kröner,et al.  Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions , 1995 .

[7]  R. Eymard,et al.  Limit boundary conditions for finite volume approximations of some physical problems , 2003 .

[8]  José Carrillo Menéndez.  Entropy solutions for nonlinear degenerate problems , 1999 .

[9]  Julien Vovelle,et al.  Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains , 2002, Numerische Mathematik.

[10]  M. Vignal.  Schemas volumes finis pour des équations elliptiques ou hyperboliques avec conditions aux limites, convergence et estimations d'erreur , 1997 .

[11]  M. Ohlberger,et al.  A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations , 2001, Numerische Mathematik.

[12]  Mario Ohlberger,et al.  A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions , 2000, Math. Comput..

[13]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[14]  Bernardo Cockburn,et al.  Convergence of the finite volume method for multidimensional conservation laws , 1995 .

[15]  Tao Tang,et al.  Error Estimates of Approximate Solutions for Nonlinear Scalar Conservation Laws , 2001 .

[16]  Claes Johnson,et al.  Adaptive finite element methods for conservation laws based on a posteriori error estimates , 1995 .

[17]  J. Carrillo.  Entropy Solutions for Nonlinear Degenerate Problems , 1999 .

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

[19]  Ralf Hartmann,et al.  Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws , 2002, SIAM J. Sci. Comput..

[20]  J. Vovelle,et al.  An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions , 2004 .

[21]  Anders Szepessy,et al.  Convergence of a streamline diffusion finite element method for a conservation law with boundary conditions , 1991 .

[22]  J. Vovelle,et al.  A Kinetic Formulation for Multidimensional Scalar Conservation Laws with Boundary Conditions and Applications , 2004, SIAM J. Math. Anal..

[23]  Claire Chainais-Hillairet,et al.  NUMERICAL BOUNDARY LAYERS FOR HYPERBOLIC SYSTEMS IN 1-D , 2001 .

[24]  J. Málek.  Weak and Measure-valued Solutions to Evolutionary PDEs , 1996 .

[25]  R. Eymard,et al.  Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes , 1998 .

[26]  Raimund Bürger,et al.  The initial-boundary value problem for a scalar conservation law , 1999 .

[27]  Jean-Paul Vila,et al.  Numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions , 1995 .

[28]  Endre Süli,et al.  A posteriori error analysis for numerical approximations of Friedrichs systems , 1999, Numerische Mathematik.

[29]  Claire Chainais-Hillairet,et al.  Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate , 1999 .

[30]  Mario Ohlberger,et al.  Adaptive Second Order Central Schemes on Unstructured Staggered Grids , 2003 .

[31]  Mario Ohlberger.  A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations , 2001 .