Kinetic functions for nonclassical shocks, entropy stability, and discrete summation by parts
arXiv:2007.08780 · doi:10.1007/s10915-021-01463-6
Abstract
We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each scheme. As established by LeFloch and collaborators, kinetic functions (for continuous or discrete models) uniquely characterize the macro-scale dynamics of small-scale dependent, undercompressive, nonclassical shock waves. We show here that various entropy-dissipative numerical schemes can yield nonclassical solutions containing classical shocks, including Fourier methods with (super-) spectral viscosity, finite difference schemes with artificial dissipation, discontinuous Galerkin schemes with or without modal filtering, and TeCNO schemes. We demonstrate numerically that entropy stability does not imply uniqueness of the limiting numerical solutions for scalar conservation laws in one space dimension, and we compute the associated kinetic functions in order to distinguish between these schemes. In addition, we design entropy-dissipative schemes for the Keyfitz-Kranzer system whose solutions are measures with delta shocks. This system illustrates the fact that entropy stability does not imply boundedness under grid refinement.
References in corpus (18)
- Review of Summation-by-parts schemes for initial-boundary-value problems
- Why many theories of shock waves are necessary. Convergence error in formally path-consistent schemes
- Relaxation Runge-Kutta Methods: Fully-Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
- Summation-by-parts operators for correction procedure via reconstruction
- Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws
- Comparison of some Entropy Conservative Numerical Fluxes for the Euler Equations
- ENO reconstruction and ENO interpolation are stable
- PyClaw: Accessible, Extensible, Scalable Tools for Wave Propagation Problems
- General Relaxation Methods for Initial-Value Problems with Application to Multistep Schemes
- Relaxation Runge-Kutta Methods for Hamiltonian Problems
- Fully-Discrete Explicit Locally Entropy-Stable Schemes for the Compressible Euler and Navier-Stokes Equations
- Shallow water equations: Split-form, entropy stable, well-balanced, and positivity preserving numerical methods
- Generalised Summation-by-Parts Operators and Variable Coefficients
- Extended Skew-Symmetric Form for Summation-by-Parts Operators and Varying Jacobians
- Stability of Correction Procedure via Reconstruction With Summation-by-Parts Operators for Burgers' Equation Using a Polynomial Chaos Approach
- Why many theories of shock waves are necessary. Kinetic functions, equivalent equations, and fourth-order models
- Entropy-stable, high-order summation-by-parts discretizations without interface penalties
- Mimetic Properties of Difference Operators: Product and Chain Rules as for Functions of Bounded Variation and Entropy Stability of Second Derivatives