Entropy stable flux correction for scalar hyperbolic conservation laws

It is known that Flux Corrected Transport algorithms can produce entropy-violating solutions of hyperbolic conservation laws. Our purpose is to design flux correction with maximal antidiffusive fluxes to obtain entropy solutions of scalar hyperbolic conservation laws. To do this we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a scheme of high-order accuracy. Flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the monotone scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution of scalar hyperbolic conservation laws. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. We present examples that show that not all numerical entropy fluxes guarantee to single out a physically correct solution of scalar hyperbolic conservation laws.

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

[2]  Philippe G. LeFloch,et al.  Fully Discrete, Entropy Conservative Schemes of ArbitraryOrder , 2002, SIAM J. Numer. Anal..

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

[4]  Eitan Tadmor,et al.  Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems , 2003, Acta Numerica.

[5]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[6]  Eitan Tadmor,et al.  The numerical viscosity of entropy stable schemes for systems of conservation laws. I , 1987 .

[7]  Eitan Tadmor,et al.  Arbitrarily High-order Accurate Entropy Stable Essentially Nonoscillatory Schemes for Systems of Conservation Laws , 2012, SIAM J. Numer. Anal..

[8]  MUSCL TYPE SCHEMES AND DISCRETE ENTROPY CONDITIONS ∗1) , 1997 .

[9]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[10]  Eitan Tadmor,et al.  PERFECT DERIVATIVES, CONSERVATIVE DIFFERENCES AND ENTROPY STABLE COMPUTATION OF HYPERBOLIC CONSERVATION LAWS , 2016 .

[11]  Stefan Turek,et al.  High-resolution FEM?FCT schemes for multidimensional conservation laws , 2004 .

[12]  V. Rusanov,et al.  The calculation of the interaction of non-stationary shock waves and obstacles , 1962 .

[13]  Dmitri Kuzmin,et al.  Explicit and implicit FEM-FCT algorithms with flux linearization , 2009, J. Comput. Phys..

[14]  Sukumar Chakravarthy,et al.  High Resolution Schemes and the Entropy Condition , 1984 .

[15]  Rainald Löhner,et al.  The design of flux-corrected transport (fct) algorithms on structured grids , 2005 .

[16]  Thomas Sonar,et al.  Entropy production in second-order three-point schemes , 1992 .

[17]  James M. Hyman,et al.  On Finite-Difference Approximations and Entropy Conditions for Shocks , 2015 .

[18]  Siddhartha Mishra,et al.  Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws , 2014, Numerische Mathematik.

[19]  Eitan Tadmor,et al.  Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes , 1984 .

[20]  Ulrik Skre Fjordholm,et al.  High-order accurate, fully discrete entropy stable schemes for scalar conservation laws , 2016 .

[21]  Stefan Turek,et al.  Flux correction tools for finite elements , 2002 .

[22]  M. Merriam.  An Entropy-Based Approach to Nonlinear Stability , 1989 .

[23]  Chi-Wang Shu,et al.  Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws , 2017, J. Comput. Phys..

[24]  S. Osher.  Riemann Solvers, the Entropy Condition, and Difference , 1984 .

[25]  Dmitri Kuzmin,et al.  Algebraic Flux Correction I. Scalar Conservation Laws , 2005 .

[26]  Philippe G. LeFloch,et al.  A fully discrete scheme for diffusive-dispersive conservation laws , 2001, Numerische Mathematik.