A Posteriori Analysis of Fully Discrete Method of Lines Discontinuous Galerkin Schemes for Systems of Conservation Laws

We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The sch...

[1]  Chi-Wang Shu,et al.  Error Estimates to Smooth Solutions of Runge-Kutta Discontinuous Galerkin Method for Symmetrizable Systems of Conservation Laws , 2006, SIAM J. Numer. Anal..

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

[3]  Christian Rohde,et al.  Error Estimates for Finite Volume Approximations of Classical Solutions for Nonlinear Systems of Hyperbolic Balance Laws , 2006, SIAM J. Numer. Anal..

[4]  Gabriella Puppo,et al.  Numerical entropy and adaptivity for finite volume schemes , 2011 .

[5]  Ricardo H. Nochetto,et al.  A posteriori error analysis for higher order dissipative methods for evolution problems , 2006, Numerische Mathematik.

[6]  B. Fornberg.  Generation of finite difference formulas on arbitrarily spaced grids , 1988 .

[7]  Elisabetta Chiodaroli,et al.  A counterexample to well-posedness of entropy solutions to the compressible Euler system , 2012, 1201.3470.

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

[9]  Arnold Lapidus.  A Detached Shock Calculation by Second Order Finite Differences , 2018 .

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

[11]  Daniela Rus,et al.  Algorithms, analysis of , 2003 .

[12]  Camillo De Lellis,et al.  On Admissibility Criteria for Weak Solutions of the Euler Equations , 2007, 0712.3288.

[13]  Chi-Wang Shu,et al.  A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes , 2006, J. Comput. Phys..

[14]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[15]  Leon Bieber,et al.  Numerical Schemes For Conservation Laws , 2016 .

[16]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[17]  Wayne H. Enright,et al.  Interpolants for Runge-Kutta formulas , 1986, TOMS.

[18]  Eberhard Bänsch,et al.  A Posteriori Error Control for Fully Discrete Crank-Nicolson Schemes , 2012, SIAM J. Numer. Anal..

[19]  Desmond J. Higham,et al.  Runge-Kutta Defect Control Using Hermite-Birkhoff Interpolation , 1991, SIAM J. Sci. Comput..

[20]  Jon Reisner,et al.  A space-time smooth artificial viscosity method for nonlinear conservation laws , 2012, J. Comput. Phys..

[21]  Charalambos Makridakis,et al.  Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic Systems of Conservation Laws , 2004, SIAM J. Numer. Anal..

[22]  Marc Laforest,et al.  An adaptive version of Glimm's scheme , 2010 .

[23]  Eleuterio F. Toro,et al.  Centred TVD schemes for hyperbolic conservation laws , 2000 .

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

[25]  C. Dafermos.  The second law of thermodynamics and stability , 1979 .

[26]  Charalambos Makridakis,et al.  Space and time reconstructions in a posteriori analysis of evolution problems , 2007 .

[27]  M. LAFOREST,et al.  A Posteriori Error Estimate for Front-Tracking: Systems of Conservation Laws , 2004, SIAM J. Math. Anal..

[28]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[29]  R. J. Diperna.  Uniqueness of Solutions to Hyperbolic Conservation Laws. , 1978 .

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

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

[32]  A. Ern,et al.  Mathematical Aspects of Discontinuous Galerkin Methods , 2011 .

[33]  Charalambos Makridakis,et al.  A Posteriori Analysis of Discontinuous Galerkin Schemes for Systems of Hyperbolic Conservation Laws , 2014, SIAM J. Numer. Anal..

[34]  Marc Laforest,et al.  An A Posteriori Error Estimate for Glimm’s Scheme , 2008 .

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