A review of a posteriori error control and adaptivity for approximations of non‐linear conservation laws

In this contribution we give an overview on recent progress in obtaining a posteriori error control for finite volume and discontinuous Galerkin approximations of non-linear hyperbolic conservation laws. The theory is based on the celebrated doubling of variables technique introduced by Kružkov (Math. USSR Sb. 1970; 10:217–243). A posteriori error control is of particular importance as it can be used for designing efficient grid adaptive schemes. The derivation of such adaptive methods is discussed and numerical experiments are given. Copyright © 2007 John Wiley & Sons, Ltd.

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

[2]  T. Gallouët,et al.  A uniqueness result for measure-valued solutions of nonlinear hyperbolic equations , 1993, Differential and Integral Equations.

[3]  Albert Cohen,et al.  Fully adaptive multiresolution finite volume schemes for conservation laws , 2003, Math. Comput..

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

[5]  Timothy J. Barth,et al.  A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems , 2000 .

[6]  Charalambos Makridakis,et al.  Finite volume relaxation schemes for multidimensional conservation laws , 2001, Math. Comput..

[7]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[8]  Endre Süli,et al.  Adaptive Finite Element Approximation of Hyperbolic Problems , 2003 .

[9]  Jérôme Jaffré,et al.  CONVERGENCE OF THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC CONSERVATION LAWS , 1995 .

[10]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[11]  Andreas Dedner,et al.  A New hp-Adaptive DG Scheme for Conservation Laws Based on Error Control , 2008 .

[12]  Bernardo Cockburn,et al.  A priori error estimates for numerical methods for scalar conservation laws. Part II : flux-splitting monotone schemes on irregular Cartesian grids , 1997, Math. Comput..

[13]  Siegfried Müller,et al.  Adaptive Multiscale Schemes for Conservation Laws , 2002, Lecture Notes in Computational Science and Engineering.

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

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

[16]  E. Süli,et al.  hp‐Discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity , 2002 .

[17]  Bernardo Cockburn,et al.  A Priori Error Estimates for Numerical Methods for Scalar Conservation Laws Part III: Multidimensional Flux-Splitting Monotone Schemes on Non-Cartesian Grids , 1998 .

[18]  Chi-Wang Shu,et al.  Error Estimates to Smooth Solutions of Runge-Kutta Discontinuous Galerkin Methods for Scalar Conservation Laws , 2004, SIAM J. Numer. Anal..

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

[20]  B. Perthame,et al.  Kruzkov's estimates for scalar conservation laws revisited , 1998 .

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

[22]  Christian Rohde,et al.  Finite‐volume schemes for Friedrichs systems in multiple space dimensions: A priori and a posteriori error estimates , 2005 .

[23]  Bernardo Cockburn,et al.  A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach , 1996, Math. Comput..

[24]  Marc Küther,et al.  Error Estimates for the Staggered Lax-Friedrichs Scheme on Unstructured Grids , 2001, SIAM J. Numer. Anal..

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

[26]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

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

[28]  A. I. Vol'pert,et al.  Cauchy's Problem for Degenerate Second Order Quasilinear Parabolic Equations , 1969 .

[29]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

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

[31]  Martin Rumpf,et al.  Towards a Unified Framework for Scientific Computing , 2005 .

[32]  Andreas Dedner,et al.  Error Control for a Class of Runge-Kutta Discontinuous Galerkin Methods for Nonlinear Conservation Laws , 2007, SIAM J. Numer. Anal..

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

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

[35]  C. Makridakis,et al.  Convergence and error estimates of relaxation schemes for multidimensional conservation laws , 1999 .

[36]  Bernardo Cockburn,et al.  Error estimates for finite element methods for scalar conservation laws , 1996 .

[37]  N. N. Kuznetsov.  Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation , 1976 .

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

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

[40]  Felix Otto,et al.  Initial-boundary value problem for a scalar conservation law , 1996 .

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

[42]  Andreas Dedner,et al.  The Distributed and Unified Numerics Environment (DUNE) , 2006 .

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

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

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

[46]  Endre Süli,et al.  hp-Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems , 2001, SIAM J. Sci. Comput..

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

[48]  Mario Ohlberger,et al.  Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method , 2005, Math. Comput..

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

[50]  A. I. Khisamutdinov.  A bounded variance estimator for computing by the Monte Carlo method the flux of particles at a point , 1976 .

[51]  Karen Dragon Devine,et al.  A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems , 2002 .

[52]  Bernardo Cockburn,et al.  An error estimate for finite volume methods for multidimensional conservation laws , 1994 .

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

[54]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .